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Primes, factors, GCF and LCM

Pythagoras arranging terracotta pots in equal rows on a sunny garden table

Identical kits, boards cut into equal pieces, buses that leave together: these word problems all come down to factors and multiples. Learn the quick tests that replace long division, how to break any number into primes, and the one question that tells you if you need the GCF or the LCM.

17 min read · 15 practice questions · 12 flashcards · Free

Factors fit inside a number; multiples grow out of it

12 chairs in 3 equal rows of 4. 3 × 4 = 12, so 3 and 4 are factors of 12, and 12 is a multiple of 3 and of 4.

A factor of a number divides it with no remainder: nothing left over. 3 is a factor of 12 because 12 ÷ 3 = 4 exactly. 5 is not a factor of 12, because 12 ÷ 5 leaves 2 over. A multiple of a number is that number times 1, 2, 3, 4 and so on. You get the multiples by counting by that number.

Factors of 121, 2, 3, 4, 6, 12
Multiples of 1212, 24, 36, 48 …
12 in primes2 × 2 × 3
Key idea

Factors are small, multiples are big

Every factor of 12 is 12 or less, and the list stops. Every multiple of 12 is 12 or more, and the list never ends. This one fact tells you which tool a word problem needs. Cutting things into equal groups gives an answer smaller than the numbers: that is a factor. Waiting for two schedules to meet again gives an answer bigger than the numbers: that is a multiple.

Next: Learn

How to solve them

  1. 1

    Find every factor: work in pairs

    Factors come in pairs that multiply to the number. Start at 1 and go up: 1 × 36, 2 × 18, 3 × 12… Test each number by dividing. When the pairs meet in the middle, you have them all.

    To list multiples, count by the number. The multiples of 9 are 9, 18, 27, 36, 45 and so on. 36 is in that list, so 36 is a multiple of 9, and 9 is a factor of 36. It is the same fact, said from the two sides.

    Factor pairs of 36
    TryDoes it divide 36?Pair
    1Yes1 × 36
    2Yes2 × 18
    3Yes3 × 12
    4Yes4 × 9
    5No: 36 ÷ 5 leaves 1 over—
    6Yes6 × 6: the pairs meet, so stop

    The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18 and 36: nine factors, read from the pairs.

    Next step: Divisibility
  2. 2

    Divisibility rules: test without dividing

    Divisible by means “divides with no remainder”. These rules let you check a big number in a few seconds, with no long division.

    Seven rules
    Divisible byCheck thisExample
    2The last digit is even: 0, 2, 4, 6 or 8358
    3The digits add up to a multiple of 3471: 4 + 7 + 1 = 12
    4The last two digits make a multiple of 41,724: 24 is 4 × 6
    5The last digit is 0 or 5635
    6It passes the rule for 2 and the rule for 3354: even, and 3 + 5 + 4 = 12
    9The digits add up to a multiple of 9738: 7 + 3 + 8 = 18
    10The last digit is 0920
    Example

    Is 5,184 divisible by 3, 4, 6 and 9?

    1. Add the digits: 5 + 1 + 8 + 4 = 18. 18 is a multiple of 3 and of 9, so 5,184 passes both rules.
    2. The last two digits are 84, and 84 is 4 × 21, so it is divisible by 4.
    3. It is even and it passes the rule for 3, so it is divisible by 6 too.

    Answer Yes, by all four

    1. 5,184 → 5 + 1 + 8 + 4
    2. 5 + 1 + 8 + 4 = 18
    3. 18 → 1 + 8 = 9
    Sum still big? Add its digits again until it is small.
    Next step: Primes
  3. 3

    Prime numbers and the factor tree

    A prime number has exactly two factors: 1 and itself. 2, 3, 5, 7, 11 and 13 are prime. A number with more than two factors, like 12, is composite. The number 1 is neither: it has only one factor. And 2 is the only even prime, because every other even number has 2 as a factor.

    Every composite number can be written as a product of primes in exactly one way, if you ignore the order. This is its prime factorization. To find it, split the number into any two factors, then keep splitting until every piece is prime. On paper this is a factor tree that branches down. Here each line is the whole number again, one split further.

    1. 180
    2. 18 × 10
    3. 2 × 9 × 2 × 5
    4. 2 × 3 × 3 × 2 × 5
    5. 2 × 2 × 3 × 3 × 5 → 2² × 3² × 5
    Split, then split again, until every piece is prime.
    Test it

    Is 69 prime?

    6 + 9 = 15, a multiple of 3, so 3 divides it: 69 is 3 × 23. Composite.

    Test it

    Is 119 prime?

    Not by 2, 3 or 5. Try 7: 7 × 10 = 70, and 119 − 70 = 49 = 7 × 7, so 119 = 7 × 17. Composite.

    Test it

    Is 53 prime?

    Not by 2, 3, 5 or 7, and the next prime is too big to check. Prime.

    Shortcut

    When to stop testing

    Test only the primes whose square is not bigger than the number. For 53: 7 × 7 = 49 is under 53, but the next prime, 11, is already too big (11 × 11 = 121). None of 2, 3, 5 and 7 divides 53, so 53 is prime.

    Next step: GCF
  4. 4

    Greatest common factor: the biggest equal groups

    A common factor divides two numbers. The greatest common factor (GCF) is the biggest one. Use it when you split two amounts into identical groups with nothing left over and want as many groups as possible.

    12 granola bars (dark squares) and 18 bottles of water (light squares) in 6 identical kits: 2 bars and 3 bottles in each. 6 is the GCF of 12 and 18.

    GCF = the primes both numbers share, multiplied

    Example

    42 bags of rice and 70 cans of beans go into identical boxes: as many boxes as possible, nothing left over. How many boxes?

    1. Write both numbers in primes: 42 is 2 × 3 × 7, and 70 is 2 × 5 × 7.
    2. They share a 2 and a 7. The GCF is 2 × 7 = 14.
    3. Each box gets 42 ÷ 14 = 3 bags of rice and 70 ÷ 14 = 5 cans.

    Answer 14 boxes

    1. GCF of 42 and 70
    2. GCF of 2 × 3 × 7 and 2 × 5 × 7
    3. shared primes: 2 and 7
    4. 2 × 7 = 14
    Write both in primes, keep what they share.
    Next step: LCM
  5. 5

    Least common multiple: when things line up again

    A common multiple is in both lists of multiples. The least common multiple (LCM) is the smallest one. Use it when two things repeat on different schedules and you want to know when they happen together again, or when you need equal amounts from packs of different sizes.

    The fast way: count by the bigger number, and stop at the first one that the smaller number divides.

    LCM = each prime, the most times it shows up in one of the numbers

    Example

    Bus A leaves every 12 minutes and bus B every 18 minutes. Both leave at 7:00 a.m. When do they next leave together?

    1. Count by the bigger number, 18: 18, 36, 54…
    2. 18 ÷ 12 leaves 6 over, so 18 does not work. 36 ÷ 12 = 3 exactly, so 36 is the LCM.
    3. They leave together again 36 minutes after 7:00, at 7:36 a.m.

    Answer 7:36 a.m.

    1. LCM of 12 and 18
    2. LCM of 2 × 2 × 3 and 2 × 3 × 3
    3. the most of each prime: 2 × 2 × 3 × 3
    4. 4 × 9 = 36
    Each prime, the most times it shows up in one number: 2 twice (from 12), 3 twice (from 18).
    Shortcut

    GCF × LCM = the two numbers multiplied

    For 12 and 18, the GCF is 6 and the LCM is 36, and 6 × 36 = 216, the same as 12 × 18. If you know three of these four numbers, you can find the fourth.

    Next step: GCF or LCM?
  6. 6

    GCF or LCM? Read what the problem is doing

    Word problems never say “find the GCF”. Ask what is happening. Is something being split into equal groups or pieces? That is the GCF, and the answer is smaller than the numbers. Is something repeating until two things match or meet? That is the LCM, and the answer is bigger than the numbers.

    “as many identical kits as possible”GCFsplit two amounts into groups
    “the longest equal pieces, nothing wasted”GCFcut lengths into pieces
    “when will they happen together again?”LCMtwo schedules meet
    “the fewest so the amounts are equal”LCMpacks of different sizes
    Splitting → GCF

    Two ropes, 32 feet and 48 feet long, are cut into equal pieces as long as possible, with nothing left. How long is each piece?

    Factors of 32: 1, 2, 4, 8, 16, 32. 16 also divides 48 (16 × 3 = 48), so the GCF is 16 feet, smaller than both ropes.

    Repeating → LCM

    One light flashes every 8 seconds and another every 14 seconds. They flash together now. When do they flash together again?

    Count by 14: 14, 28, 42, 56. 56 ÷ 8 = 7, so 8 divides 56 and the LCM is 56 seconds, bigger than both.

    Packs → LCM

    Hot dogs come 10 to a pack and buns 8 to a pack. What is the fewest hot dogs you can buy to have one bun for each?

    The LCM of 10 and 8 is 40: 4 packs of hot dogs and 5 packs of buns.

    Next step: Three examples
  7. 7

    Three worked examples, from easy to test level

    Each one starts with the first try many people make, crossed out, and then the way that works. Read the wrong try too: it is the kind of answer the test puts in the choices.

    The test-level one hides a second step. The GCF tells you how many packages there are, but the question asks what goes inside one. Read the last sentence again before you choose.

    Example · Easy

    What is the LCM of 6 and 14?

    First try6 × 14 = 84

    1. 84 is a common multiple, because 6 and 14 both divide it. But the question asks for the least one, and there may be a smaller one.
    2. Count by the bigger number, 14: 14, 28, 42…
    3. 14 ÷ 6 and 28 ÷ 6 leave remainders. 42 ÷ 6 = 7 exactly.
    4. The LCM is 42.

    Answer 42

    Example · Medium

    Write 252 as a product of primes.

    First try252 = 4 × 63 = 4 × 7 × 9

    1. 4 and 9 are not prime, so the first try stopped too early. Keep splitting until every piece is prime.
    2. 252 is even: 252 is 2 × 126. 126 is even too: 126 is 2 × 63.
    3. 6 + 3 = 9, so 3 divides 63: 63 is 3 × 21, and 21 is 3 × 7.
    4. All the pieces together: 2 × 2 × 3 × 3 × 7, which is 2² × 3² × 7.

    Answer 2² × 3² × 7

    Example · Test level

    A supply sergeant has 48 bottles of water and 80 energy bars. She packs them into identical care packages, as many packages as possible, with nothing left over. How many items are in each package?

    First tryThe GCF of 48 and 80 is 16, so 16 items

    1. 16 is the right GCF, but it answers a different question: it is the number of packages, not what goes inside one.
    2. In primes, 48 is 2 × 2 × 2 × 2 × 3 and 80 is 2 × 2 × 2 × 2 × 5. They share four 2s, and 2 × 2 × 2 × 2 is 16, so there are 16 packages.
    3. Water in each package: 48 ÷ 16 = 3 bottles. Bars in each package: 80 ÷ 16 = 5 bars.
    4. Items in each package: 3 + 5 = 8.

    Answer 8 items

Next: Watch out

4 traps in the answer choices

The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.

Trap

Multiplying to get the LCM

The LCM of 6 and 9:

Wrong6 × 9 = 54

Right18, the first multiple of 9 that 6 divides

Trap

GCF and LCM mixed up

Boards of 30 and 45 inches cut into the longest equal pieces, nothing wasted:

WrongLCM: pieces of 90 inches

RightGCF: pieces of 15 inches

Trap

A composite left in the answer

The prime factorization of 72:

Wrong8 × 9

Right2 × 2 × 2 × 3 × 3

Trap

Odd does not mean prime

Is 87 prime?

WrongIt is odd, so it is prime

Right8 + 7 = 15, so 3 divides it: 87 is 3 × 29

Next: Practice

15 questions in three levels

Warm-up first, then core, then test level. Pick an answer, then read why.

On test day

Factors and multiples are behind many word problems: identical kits, boards cut into pieces, buses and shifts that repeat. On the computer test, Arithmetic Reasoning has 15 questions in 55 minutes and Mathematics Knowledge has 15 in 31 minutes. On paper, Arithmetic Reasoning is 30 questions in 36 minutes and Mathematics Knowledge is 25 in 24 minutes.

No calculator. The divisibility rules replace long division. And when a question asks for a least common multiple, watch for the two numbers simply multiplied together among the choices: it is a common multiple, but often not the least.

A tip: before you calculate, decide if the answer should be smaller or bigger than the numbers in the problem. Splitting into groups gives smaller (GCF); waiting for things to meet gives bigger (LCM). On the computer test you cannot go back to a question, so make that check every time.

Question 1 of 15 · Warm-up

Which number is a factor of 56?

Show the answer

D. A factor divides 56 with no remainder. 8 × 7 = 56, so 8 is a factor. 3 fails the digit test: 5 + 6 = 11, not a multiple of 3. 5 fails because 56 does not end in 0 or 5. 6 fails too: 6 × 9 = 54 and 6 × 10 = 60, so counting by 6 jumps over 56. The factor is 8.

Question 2 of 15 · Warm-up

Which of these numbers is prime?

Show the answer

A. Test each one with small primes. 51: 5 + 1 = 6, so 3 divides it (51 is 3 × 17). 57: 5 + 7 = 12, so 3 divides it too (57 is 3 × 19). 91 looks prime, but 7 × 13 = 70 + 21 = 91. 41 is not divisible by 2, 3 or 5, and 7 × 7 = 49 is already bigger than 41, so you can stop. The prime is 41.

Question 3 of 15 · Warm-up

Which number is divisible by 4?

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C. Look only at the last two digits. 54 ÷ 4 leaves 2 over, 18 ÷ 4 leaves 2 over, and 02 is just 2, which 4 does not divide. 16 is 4 × 4, so 3,716 is divisible by 4. All four numbers are even, but being even only proves they divide by 2. The answer is 3,716.

Question 4 of 15 · Warm-up

What is the greatest common factor of 18 and 27?

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A. List the factors of the smaller number, 18: 1, 2, 3, 6, 9, 18. The ones that also divide 27 are 1, 3 and 9, and the biggest is 9. 18 does not divide 27, 27 is too big to divide 18, and 54 is the LCM, the smallest number they both divide. The GCF is 9.

Question 5 of 15 · Warm-up

What is the least common multiple of 9 and 12?

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B. Count by the bigger number, 12: 12, 24, 36. 12 ÷ 9 and 24 ÷ 9 leave remainders, but 36 ÷ 9 = 4 exactly. 3 is the GCF, not the LCM. 108 is 9 × 12, a common multiple but not the least, and 72 is common too but bigger than 36. The LCM is 36.

Question 6 of 15 · Core

Which is the prime factorization of 90?

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B. Split until every piece is prime. 90 is 9 × 10, 9 is 3 × 3 and 10 is 2 × 5. So 90 is 2 × 3 × 3 × 5, which is 2 × 3² × 5. The other choices also multiply to 90, but each keeps a piece that is not prime: 15, 9 and 30. The answer is 2 × 3² × 5.

Question 7 of 15 · Core

Which number is divisible by 6?

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C. Divisible by 6 means it passes two tests: it is even (rule for 2) and its digits add up to a multiple of 3 (rule for 3). 3,315 passes the test for 3 (3 + 3 + 1 + 5 = 12), but it is odd. 4,522 and 6,124 are even, but their digits add up to 13, not a multiple of 3. 5,418 is even and 5 + 4 + 1 + 8 = 18. The answer is 5,418.

Question 8 of 15 · Core

What is the greatest common factor of 54 and 90?

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B. Write both in primes: 54 is 2 × 3 × 3 × 3 and 90 is 2 × 3 × 3 × 5. They share 2 × 3 × 3, which is 18. 9 is a common factor, but not the greatest: it leaves out the shared 2. 27 divides 54 but not 90. 270 is the LCM. The GCF is 18.

Question 9 of 15 · Core

What is the least common multiple of 8 and 20?

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D. Count by the bigger number, 20: 20, 40. 20 ÷ 8 leaves 4 over, so 20 is not a multiple of 8. 40 ÷ 8 = 5 exactly. 4 is the GCF, and 28 is 8 + 20, an addition instead of a multiple. The LCM is 40.

Question 10 of 15 · Core

Pencils come in packs of 12 and erasers in packs of 8. Mr. Diaz wants the same number of pencils and erasers. What is the fewest packs of pencils he can buy?

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A. Equal amounts from packs of different sizes is an LCM problem. Count by 12: 12, 24. 24 ÷ 8 = 3, so 24 pencils and 24 erasers is the smallest match. 24 pencils is 24 ÷ 12 = 2 packs. 3 is the packs of erasers, 4 is the GCF of 12 and 8, and 24 is the number of pencils, not packs. The fewest packs of pencils: 2.

Question 11 of 15 · Test level

A shuttle leaves the main gate every 15 minutes and a second shuttle leaves every 25 minutes. Both leave together at 8:00 a.m. How many times do they leave together from 8:00 a.m. to 1:00 p.m., counting both of those times?

Show the answer

B. They leave together again after the LCM of 15 and 25. Count by 25: 25, 50, 75. 75 ÷ 15 = 5, so the LCM is 75 minutes, which is 1 hour 15 minutes. Starting at 8:00: 8:00, 9:15, 10:30, 11:45 and 1:00. 4 forgets to count 8:00. 21 counts every 15 minutes and 13 every 25 minutes: each is the schedule of one shuttle only. They leave together 5 times.

Question 12 of 15 · Test level

A medic has 45 bandages and 60 gauze pads. She packs them into identical first-aid kits, as many kits as possible, with nothing left over. How many items are in each kit?

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C. Splitting into identical groups is a GCF problem. 45 is 3 × 3 × 5 and 60 is 2 × 2 × 3 × 5. They share a 3 and a 5, and 3 × 5 = 15, so there are 15 kits. Each kit gets 45 ÷ 15 = 3 bandages and 60 ÷ 15 = 4 gauze pads. 3 and 4 count only one kind of item, and 15 is the number of kits, not what is inside one. Each kit holds 3 + 4 = 7.

Question 13 of 15 · Test level

The GCF of two numbers is 6 and their LCM is 72. One of the numbers is 18. What is the other number?

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C. Use GCF × LCM = the two numbers multiplied. 6 × 72 is 6 × 70 = 420 plus 6 × 2 = 12, so 432. Then 18 times the other number is 432. Halve both: 216 ÷ 9. 9 × 20 = 180, and 216 − 180 = 36 = 9 × 4, so 216 ÷ 9 = 24. Check: 24 and 18 share only 2 × 3 = 6, and 24 × 3 = 72 is the first multiple of 24 that 18 divides. 4 is 72 ÷ 18, 12 is 72 ÷ 6, and 54 is 72 − 18. The other number is 24.

Question 14 of 15 · Test level

Three warning lights blink every 4, 6 and 9 seconds. They blink together right now. In how many seconds will all three blink together again?

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D. Find the LCM of all three. Count by the biggest, 9: 9, 18, 27, 36. 18 works for 6 and 9, but 18 ÷ 4 leaves 2 over. 27 is odd, so 4 and 6 cannot divide it. 36 ÷ 4 = 9 and 36 ÷ 6 = 6, so 36 works for all three. 12 and 24 work only for 4 and 6, and 18 only for 6 and 9. All three blink together after 36 seconds.

Question 15 of 15 · Test level

A group of recruits can be split into teams of 4, 6 or 10 with nobody left over. What is the smallest number of recruits the group could have?

Show the answer

D. The number must be a multiple of 4, of 6 and of 10: that is the LCM. In primes, 4 is 2 × 2, 6 is 2 × 3 and 10 is 2 × 5. Take each prime the most times it appears: 2 × 2 × 3 × 5, which is 60. 20 works for 4 and 10 but not for 6. 30 works for 6 and 10 but not for 4. 40 works for 4 and 10 but not for 6. The smallest group has 60 recruits.

Next: Flashcards

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