Factors, Multiples, Primes, Squares and Cubes

You have 18 counters. Can you split them into equal rows? A factor tells you a row count that works with nothing left over.

Divide to check a factor

Try 3 rows. Each row gets 6 counters:

The division is exact, so 3 is a factor of 18. Try 4 rows instead. You can place 4 counters in each row, but 2 remain:

That leftover 2 means 4 is not a factor of 18. A remainder is what is left after making equal whole groups.

Find the partner

A multiplication fact gives two factors at once:

Both 3 and 6 are factors of 18. Together they make a factor pair. You can read the fact as 3 rows of 6 or 6 rows of 3.

Lay the counters in a rectangle, with every row the same length. This arrangement is an array. For 18 counters, the different rectangles show these pairs:

• 1 and 18
• 2 and 9
• 3 and 6

Turning a rectangle does not give a new factor pair. The pair 6 and 3 repeats the pair 3 and 6. When you list pairs, write each one just once.

Read the full factor list

Take both numbers from every pair. Put them in order:

1, 2, 3, 6, 9, 18.

Each number in this list divides 18 exactly. Notice the ends: the smallest factor is 1, and the largest is 18. Every counting number has itself and 1 as factors. For 1, these are the same number, so its only factor is 1.

A factor of a counting number cannot be larger than that number. You cannot split 18 counters into 36 nonempty equal groups. But 18 is a factor of 36, since two groups of 18 make 36.

Listen for words with the same meaning

These sentences all tell you the same fact:

• "3 is a factor of 18."
• "3 is a divisor of 18."
• "18 is divisible by 3."

They all say that division by 3 leaves no remainder. Divisor is another name for a factor in this setting.

Adding two numbers to make 18 does not make them factors. For example, 8 and 10 add to 18, but neither divides 18 exactly. Use multiplication or division to check a factor.

Guessing factor pairs can leave gaps. Start with 1 and test the next whole number each time. A neat record helps you find every pair.

Search in order

Find the factor pairs of 40. For each test, divide 40 by the number. An exact answer gives its partner.

The next test would be 7. But two numbers that are both at least 7 cannot multiply to 40:

So the pairs are complete. Any larger factor must partner with a smaller one you have already tested.

You can also stop when you reach a partner already recorded. For 20, you find 1 and 20, 2 and 10, then 4 and 5. Reaching 5 would only turn around the last pair.

Turn pairs into a factor list

For 40, collect both members of every pair. Write each factor once, from smallest to largest:

1, 2, 4, 5, 8, 10, 20, 40.

Check by multiplying each pair. In a factor rainbow, the smallest and largest factors join. The next two join inside them. Each arch shows a pair with the same product.

Watch for a factor paired with itself

Try 36. The pairs are:

Test 5 too; it leaves a remainder. At 6 and 6, the partners meet. Write 6 just once in the factor list:

1, 2, 3, 4, 6, 9, 12, 18, 36.

There are nine factors, not ten. A number with a self-pair is a square number. Its factors come in pairs except for that one middle factor. That is why square numbers have an odd number of factors.

Know when a short list is complete

For 13, the pair 1 and 13 works. Tests with 2 and 3 leave remainders. There is no need to test 4 because its square is already larger than 13. So 13 has only two factors. A number greater than 1 with only two factors is prime.

Do not stop just because a test fails. For 40, failing at 3 does not let you skip 4 or 5. Keep testing in order until the partners meet or the tested number's square would be too large.

A multiple is a total you get by multiplying a number by a whole number. Think of building equal groups instead of splitting a total.

Build a list of multiples

Each bag holds 8 beads. One bag holds 8 beads, two hold 16, and three hold 24. Keep adding 8:

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, …

This is skip counting. The dots show that the list continues. A multiplication table does not end after ten groups. Eleven groups of 8 make 88, and twelve groups make 96.

You can skip count by larger amounts too. Counting by 25 gives 25, 50, 75, 100, 125. Counting by 1,000 gives 1,000, 2,000, 3,000, 4,000.

There is always a next multiple: add the same number again. So the multiples of a positive number go on forever. Its factor list ends, because its factors cannot be larger than itself.

Read one fact in two ways

Look at this fact:

• 8 and 9 are factors of 72.
• 72 is a multiple of 8.
• 72 is also a multiple of 9.

The smaller numbers build the total. You can name either side of the relationship, but keep the words in the right order. Say, "8 is a factor of 72," or, "72 is a multiple of 8."

Test a possible multiple

Is 78 a multiple of 6? Divide:

There is no remainder, so 13 groups of 6 make 78. That makes 78 a multiple of 6 and 6 a factor of 78.

For 79, the answer is different:

One is left over. You cannot make exactly 79 using whole groups of 6.

Use patterns with care

Multiples of 2 are even. Multiples of 5 end in 0 or 5. Multiples of 10 end in 0. These patterns let you rule out many numbers quickly.

Every multiple of 6 is also a multiple of 2 and 3. A group of 6 can split into three 2s or two 3s. But a multiple of 2 need not be a multiple of 6: 10 is a multiple of 2 and leaves a remainder when divided by 6.

Our lists begin with one group, so the first listed multiple of 8 is 8. Zero is also a multiple, since multiplying by zero gives zero. It is left out when we list the positive multiples. The size check "the number itself or larger" applies to those positive multiples.

A divisibility rule tells you whether division will be exact. Some rules need only the ending of a number, even when the number is very large.

Check the ones digit

Use these three rules:

For 6,284, the last digit is 4. It passes the test for 2, so it is even. It fails the tests for 5 and 10.

For 3,715, the last digit is 5. It passes the test for 5. It is odd, so it fails the test for 2. It also fails the test for 10.

A number ending in 0 passes all three tests. For example, 8,430 is divisible by 2, 5, and 10.

Why the last digit is enough

Split 6,284 into tens and leftover ones:

The 6,280 part is made of whole tens. Every ten splits exactly into five 2s, two 5s, or one 10. So that part passes all three tests. Only the leftover 4 can leave a remainder. Four splits into 2s but not into 5s or 10s.

The beginning of the number does not decide. A number can start with 5 and fail the test for 5. For example, 53 ends in 3, so it fails.

Check hundreds and thousands

For 100, check the last two digits. They must be 00. For 1,000, check the last three. They must be 000.

• 27,600 is divisible by 100 but not by 1,000.
• 27,060 is not divisible by 100: its ending is 60.
• 27,000 is divisible by both 100 and 1,000.

These endings show that the number contains whole hundreds or whole thousands, with no smaller part left over. You can check 27,600 by division:

A zero somewhere in the number is not enough. It must have the right number of zeros at the end.

Link the rules

A number divisible by 10 is divisible by 2 and 5. Passing just one of those tests is not enough for 10. The number 45 passes the test for 5, but it ends in 5, not 0.

Passing both tests does work: a number ending in 0 or 5 must end in 0 to be even as well. That gives the ending needed for divisibility by 10.

For 3 and 9, the last digit is not enough. Add every digit. This total is the digit sum, and it gives you two useful tests.

Use a digit sum for 3

Is 2,754 divisible by 3? Add its digits:

Eighteen is divisible by 3, so 2,754 is too. You do not need to divide the whole number to answer yes.

Now test 2,755. Its digit sum is 19. Nineteen is not divisible by 3, so 2,755 is not divisible by 3.

If a digit sum is still large, add its digits again. A sum of 39 becomes 12, then 3. This keeps the same yes-or-no result for divisibility by 3.

Use the same sum for 9

A number is divisible by 9 when its digit sum is divisible by 9. The sum 18 passes, so 2,754 is divisible by 9 as well as by 3.

For 2,751, the sum is 15. Fifteen is divisible by 3 but not by 9. So 2,751 passes the test for 3 and fails the test for 9.

Every multiple of 9 is a multiple of 3. The reverse does not hold: three groups of 5 make 15, which is not a whole number of 9s.

Why adding digits works

A ten is one more than 9. A hundred is one more than 99. A thousand is one more than 999. Each place-value piece can leave its digit behind after taking away groups of 9.

For 243, split the hundreds and tens like this:

The first two parts divide exactly by 9, and also by 3. Only the digit sum in parentheses can leave anything over. That is why the sum decides both tests.

Make two checks for 6

A number is divisible by 6 when it is divisible by both 2 and 3. Check that it is even, then check its digit sum.

For 426:

1. It ends in 6, so it is even.
2. Its digit sum is 12, which is divisible by 3.
3. Both checks pass, so 426 is divisible by 6.

The number 423 passes the digit-sum test, but it is odd. It fails for 6. The number 424 is even, but its digit sum is 10. It fails for 6 too.

Do not use only the last digit for 3. The number 16 ends in 6 but has digit sum 7, so it fails. The number 24 ends in 4 but has digit sum 6, so it passes.

Some divisibility tests need more than one final digit. You can still ignore most of a long number. #### Test the last two digits for 4

A number is divisible by 4 when the number made by its last two digits is divisible by 4.

For 6,932, test 32:

So 6,932 is divisible by 4. For 6,934, the ending 34 leaves a remainder, so the whole number fails.

Why can you ignore the rest? Every hundred contains exactly 25 groups of 4. The 6,900 part already divides by 4. Only the final 32 matters.

The number 34 is even but is not a multiple of 4. Another check is to halve twice. Both answers must be whole numbers. For 32, the halves are 16 and then 8.

Test the last three digits for 8

Every thousand contains exactly 125 groups of 8. So use the last three digits for the test for 8.

For 4,216, test 216:

So 4,216 passes. You can halve 216 three times instead: 108, then 54, then 27. Every result is whole.

For a number with three digits or fewer, test the whole number. Watch the hundreds digit: 1,112 and 1,212 both end in 12, but only 1,112 is divisible by 8.

Passing the tests for 2 and 4 is not enough for 8. Every multiple of 4 already passes for 2. Checking both repeats information. The number 20 passes both but does not divide exactly by 8.

Recognize the endings for 25 and 50

You can use these short lists:

For 25, four equal jumps fill each hundred. For 50, two equal jumps fill each hundred. So the hundreds part always works, and the ending decides.

The number 8,275 ends in 75. It is divisible by 25, but not by 50. The number 8,250 ends in 50, so it passes both tests.

Match the test to the divisor

Use the ending for 4 or 8, not the digit sum. Use the digit sum for 3 or 9, not just the ending. For 6, combine the tests for 2 and 3.

To test 7, use division. For example:

So 98 is divisible by 7. A remainder would mean it is not.

A common factor divides two numbers exactly. The word "common" means shared. You need a number that appears in both factor lists.

Compare complete lists

Find the common factors of 16 and 24. First use factor pairs to build each list.

• For 16: 1 and 16, 2 and 8, 4 and 4.
• For 24: 1 and 24, 2 and 12, 3 and 8, 4 and 6.

Now write the factors in order:

The numbers in both lists are 1, 2, 4, and 8. Those are the common factors.

The number 6 is a factor of 24 but not of 16. That makes it a factor of just one number, not a common factor. Check both numbers every time.

Put shared factors in the overlap

A Venn diagram has overlapping circles. Label one circle "Factors of 16" and the other "Factors of 24."

Put 1, 2, 4, and 8 in the overlap. Write each just once. The overlap belongs to both circles. Put 16 in the 16-only part. Put 3, 6, 12, and 24 in the 24-only part.

A number such as 5 goes outside both circles because it divides neither number exactly. If you are sorting only the factors from your two lists, you will have no numbers to place outside.

Find the greatest shared factor

The greatest common factor, or GCF, is the largest number in the shared list. For 16 and 24, it is 8. You may also hear "highest common factor," or HCF, for the same idea.

Any two counting numbers share the factor 1. Some share nothing else. For example:

• Factors of 9: 1, 3, 9.
• Factors of 14: 1, 2, 7, 14.

Their only common factor is 1, so their GCF is 1.

Check what a shared factor means

Imagine 16 red beads and 24 blue beads. You want the same number of equal groups for each color, with no beads left. You can use 1, 2, 4, or 8 groups because each of those numbers divides both amounts exactly.

For 4 groups, each group gets 4 red beads and 6 blue beads:

The group sizes for the two colors need not match. What matches here is the number of groups. No common factor can be larger than the smaller total, since it must divide that total too.

A common multiple is a total that both numbers can make. It appears in both lists of positive multiples.

Look for matching totals

Find the common multiples of 4 and 6. Start each list at the number itself:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, …
• Multiples of 6: 6, 12, 18, 24, 30, 36, …

Both lists contain 12, 24, and 36. These are their first three common multiples. Check 24 in both ways:

Both divisions are exact. A total must work for both numbers, not just one.

Find the first match

The smallest positive common multiple is the least common multiple, or LCM. It is also called the lowest common multiple. For 4 and 6, the LCM is 12.

Multiplying 4 and 6 gives 24. That is a common multiple, but it misses the earlier match at 12. Multiplying two numbers always gives a common multiple; you still need to check whether there is a smaller one.

To shorten your search, list multiples of the larger number. Then test each with the smaller number. For 8 and 12:

1. Start at 12. It does not divide exactly by 8.
2. Try 24. It divides exactly by 8.
3. So 24 is the first shared multiple.

You tested the possible totals in order, so you did not skip a smaller match.

Keep the shared pattern going

Once you find the first shared multiple, keep adding it. The common multiples of 8 and 12 begin 24, 48, 72, 96. They continue forever. There is no largest common multiple.

When one number is already a multiple of the other, start with that larger number. For 5 and 15, the first match is 15. Every multiple of 15 is also a multiple of 5, so the shared list is 15, 30, 45, 60, …

Keep factors and multiples straight

A common factor divides into both numbers. A common multiple can be divided by both numbers. For 4 and 6, the common factor list is 1, 2. The common multiple list starts at 12.

Picture two sets of jumps along a number line. Start both at 0. One jumps 4 spaces each time; the other jumps 6. Mark the landings. The positive places both sets reach are 12, 24, 36, and so on. The jumps help you see why the matches continue past the end of a short written list.

You can sort a counting number by counting its factors. Prime and composite tell you whether the number has just two factors or more than two.

Exactly two factors means prime

A prime number is greater than 1 and has exactly two factors: 1 and itself. For 11, the only pair is 1 and 11. So its factors are 1 and 11, and it is prime.

The prime numbers up to 20 are:

2, 3, 5, 7, 11, 13, 17, 19.

Each has exactly two different factors. You can arrange a prime number of tiles in only a single-row rectangle, allowing that rectangle to turn.

More than two factors means composite

A composite number is greater than 1 and has more than two factors. The factors of 10 are 1, 2, 5, and 10. It has four factors, so it is composite.

You do not need the whole factor list to show a number is composite. Find one pair other than 1 and the number itself:

The factors 3 and 7 show that 21 is composite. Tiles can make a 3-by-7 rectangle as well as a 1-by-21 rectangle.

A self-pair counts too. For 9, the factors are 1, 3, and 9. There are three different factors, so 9 is composite.

Give 1 its own place

The number 1 has just one factor: 1. Writing 1 twice does not give it two different factors. So 1 is neither prime nor composite. Zero is neither prime nor composite either.

Every whole number greater than 1 fits exactly one of the two groups: prime or composite.

Why 2 is special

The factors of 2 are 1 and 2. That makes 2 prime. Every larger even number also has 2 as a factor, along with 1 and itself. So every even number greater than 2 is composite.

That leaves 2 as the only even prime number. Do not cross it out just because it is even.

Odd numbers need checking. Some are prime, such as 13. Others are composite, such as 15 or 35:

Being odd only tells you that 2 is not a factor. It says nothing about factors such as 3, 5, or 7.

Make your reason visible

For a composite number, give a factor pair that proves it. For a prime number, explain that its only factors are 1 and itself. "It looks prime" is not a factor check. A number such as 51 may look unfamiliar, but its digit sum is 6, so 3 is a factor and it is composite.

A sieve separates things. The Sieve of Eratosthenes separates primes from composite numbers by crossing out multiples on a 1–100 grid.

Work across the grid

1. Cross out 1. It is not prime.
2. Circle 2. Cross out every other multiple of 2.
3. Circle 3. Cross out its other multiples.
4. Circle 5. Cross out its other multiples.
5. Circle 7. Cross out its other multiples.
6. Circle every number still left. They are all prime.

Keep each prime itself. When you circle 5, start crossing out at 10, not 5. A composite number may already be crossed out by an earlier step. It stays crossed out.

Why stop after 7?

Imagine a composite number up to 100. It has a factor pair with both factors greater than 1. If both were at least 11, the product would be at least 121. That is too large.

So the smaller factor is at most 10. The numbers from 2 to 10 are built from multiples of 2, 3, 5, or 7. Crossing out those four sets of multiples catches every composite number up to 100.

This stopping rule depends on the upper limit 100. It lets you check the whole grid without crossing out multiples of every prime you find.

Read the primes

There are 25 primes up to 100:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Learn the primes up to 19 so you can recall them quickly. For larger numbers up to 100, the tests for 2, 3, 5, and 7 can settle the question.

Test one number without a grid

Is 83 prime?

• It is odd, so 2 is not a factor.
• Its digit sum is 11, so 3 is not a factor.
• It does not end in 0 or 5, so 5 is not a factor.
• Dividing by 7 gives 11 remainder 6, so 7 is not a factor.

None of the four divides 83. Since 83 is between 11 and 100, it is prime.

Now try 77. It passes the first three checks, but:

So 77 is composite. Do not stop after checking only 2, 3, and 5.

Primes greater than 5 end in 1, 3, 7, or 9. Those endings help you find possible primes, but they do not settle the test. For example, 49 ends in 9 and is composite because it is 7 squared.

Prime numbers keep appearing beyond 100. Their gaps vary, and they never run out.

A square number comes from multiplying a whole number by itself. You can see why it has that name by making a square of dots.

Read the small raised 2

Five rows of five dots make 25 dots:

Read the small raised 2 as "squared." It tells you to use two copies of 5 as factors. It does not mean two groups of 5. Those would total only 10.

Here are the squares from 1 squared to 12 squared:

Grow a square

The list starts 1, 4, 9, 16, 25, 36. The gaps are 3, 5, 7, 9, 11: odd numbers in order.

To grow a 5-by-5 square into a 6-by-6 square, add a row of 5, a column of 5, and one corner dot. That adds 11 dots:

The gap grows as the square grows. You cannot get every next square by adding the same amount.

Work backward with a square root

The square root of a square number tells you its side length in dots. The square root of 49 is 7 because 7 rows of 7 make 49:

The sign √ means "square root." Squaring starts with the side and finds the total. Taking the square root starts with the total and finds the side.

For example, a square picture uses 121 tiny tiles. Since 11 squared is 121, it has 11 tiles along each side. Half of 121 would not tell you the side length.

Decide whether a number is square

The number 50 is between 49 and 64. These are consecutive squares: 7 squared and 8 squared. No whole number between 7 and 8 can give another whole-number side length. So 50 is not square and has no whole-number square root.

Squares can be odd or even. Squaring an odd number gives an odd square; squaring an even number gives an even square. Also, 1 squared is still 1.

A positive square number has a factor paired with itself. For 16, it is 4 and 4. The factors 1, 2, 4, 8, and 16 total five. That unpaired middle factor makes the factor count odd. Every square number greater than 1 is composite, since its side length is a factor between 1 and the total.

A cube has equal length, width, and height. A cube number counts the small blocks in a cube with a whole number of blocks along each edge.

Read the small raised 3

A cube 3 blocks long, 3 wide, and 3 high contains:

Read this as "3 cubed." The raised 3 tells you to multiply three copies of 3. It does not mean multiply 3 by 3 just once, or add three 3s.

Imagine one layer first. It has 3 rows of 3 blocks, so it contains 9. There are 3 equal layers. Three layers of 9 make 27.

Multiply in two steps

To find 5 cubed, first find one square layer:

Then multiply by the number of layers:

The cube contains 125 small blocks. Keep the starting number the same for all three factors. The number 25 is an intermediate total, not a new edge length.

Build the cube list

Learn these cubes: 1, 8, 27, 64, 125, and 1,000. They come from 1, 2, 3, 4, 5, and 10 cubed.

You can work out the cubes between 5 and 10 with the same two steps:

For example, 7 cubed means 49 blocks per layer and 7 layers. Multiplying 49 by 7 gives 343.

Compare a square and a cube

For 4 squared, use two factors of 4. For 4 cubed, use three:

A square array has rows and columns. A cube has those rows and columns in every layer, with a stack of equal layers. That extra factor makes cubes grow faster. At 10, the square is 100 and the cube is 1,000.

Cubes can be odd or even: 3 cubed is odd, while 4 cubed is even.

Some numbers can be made both ways. The number 64 is 8 squared and 4 cubed. So "square" and "cube" are not separate boxes that can never overlap. Up to 1,000, the positive numbers in both lists are 1, 64, and 729. For 729, you can check 27 times 27 or 9 times 9 times 9.

A number problem may ask you to split a total or to build one. Read what the groups mean before choosing factors or multiples.

Split a total with factors

A class has 42 students. They form equal teams with more than 3 and fewer than 9 students per team. What team sizes work?

Start with factor pairs:

The factors are 1, 2, 3, 6, 7, 14, 21, and 42. The extra clues keep only 6 and 7. Six students per team gives 7 teams. Seven students per team gives 6 teams.

Both arrangements use every student. Check the exact words: "more than 3" leaves out 3, and "fewer than 9" leaves out 9.

Decide whether turning matters

With 18 chairs, the pair 3 and 6 gives one factor pair. But a room can have 3 rows of 6 chairs or 6 rows of 3 chairs. Those are different row layouts.

The full pairs for 18 are 1 and 18, 2 and 9, 3 and 6. Each pair gives two row layouts, for six layouts in all. Always check whether the question asks for factor pairs or for rows facing the same direction.

Build totals with multiples

Glue sticks come in packs of 7. Whole packs give totals of 7, 14, 21, 28, and so on. Can you buy exactly 63?

Yes: 9 packs give exactly 63 sticks. To buy 65, whole packs do not work. Nine packs give 63 and ten give 70.

A remainder also matters when sharing. If 46 cards are shared among 5 children, each gets 9 and 1 card remains. So 5 is not a factor of 46, and the sharing is not exact.

Check a calculation with a rule

Someone writes, "6 × 38 = 226." Every multiple of 6 must pass the test for 3. The digit sum of 226 is 10, so this answer fails. Recalculate:

The new digit sum is 12, so it passes that test. A quick test can catch an impossible answer. To know the exact answer, still do the multiplication: several totals can pass the same test.

Use the shape as a clue

A square mosaic has 81 tiles. Its side has 9 tiles because 9 squared is 81. A cube with 5 blocks along every edge has 25 blocks per layer and 5 layers, giving 125 blocks.

Name what your answer counts: students per team, teams, packs, tiles per side, or blocks. The same multiplication fact can answer different questions.

A number can have several properties at once. Sorting diagrams and clue lists help you keep track of them without guessing.

Sort with two circles

Draw a Venn diagram with circles labeled "Even" and "Multiples of 5." Sort these numbers: 4, 5, 10, 13, 15, 20.

• Even only: 4.
• Multiples of 5 only: 5, 15.
• Both, in the overlap: 10, 20.
• Neither, outside both circles: 13.

Check each number against both labels. A number belongs in the overlap only when both properties hold.

Sort with a grid

A Carroll diagram uses a row label and a column label for each box. Here is a way to sort 1 to 10:

The "not prime" column includes 1 even though 1 is not composite. A label saying "not prime" means everything that fails the prime test.

Work through number clues

Try this puzzle: "I am a two-digit square number. My digits add to 9. I am even."

Start with two-digit squares: 16, 25, 36, 49, 64, 81. Check the digit sums. Only 36 and 81 have sum 9. Now use the final clue: 36 is even, and 81 is odd. The number is 36.

A short list makes it easy to see what each clue removes. Check your final number against every clue, including the first one.

Test words like always

"A multiple of 4 is even" is always true. Every group of 4 can split into two groups of 2, so any whole number of 4s can split into 2s.

"A square number is odd" is sometimes true. Nine is an odd square; 16 is an even square. You need one example that works and one that does not.

"A prime number greater than 2 is even" is never true. Every even number greater than 2 has the extra factor 2, so it is composite.

Break a claim with one example

A counterexample shows that an "always" claim is false. For "All odd numbers are prime," 9 works as a counterexample because it is odd and has the factor 3.

Finding several odd primes cannot settle that claim. You need a reason that covers every case to show "always."

Factors can help solve unusual clues too. The numbers below 50 with exactly three factors are 4, 9, 25, and 49. Each is a prime squared. Its only factors are 1, that prime, and the square itself. For 49, these are 1, 7, and 49.