Factors and Multiples

Study Sheet

Factors and Multiples

What divides evenly

Factors in Pairs

Concept
The Big Idea
1234612
Each arc joins a factor pair of 12: 1 × 12, 2 × 6, 3 × 4.

Factors come in pairs that multiply to the number, which is why you only have to test up to the square root — after that the pairs repeat.

A factor of 1212 is a number that divides it exactly: 1,2,3,4,6,121, 2, 3, 4, 6, 12. A multiple of 1212 is 1212 times something: 12,24,36,…12, 24, 36, \ldots Factors are small and finite; multiples go on forever. A prime number has exactly two factors, itself and 11.

Example
The factors of 12
1 × 122 × 63 × 4
Every way to arrange 12 dots in equal rows is a factor pair.

1×121 \times 12, 2×62 \times 6, 3×43 \times 4. So the factors are 1,2,3,4,6,121, 2, 3, 4, 6, 12 — six of them. Each pair is one possible array.

Example
The factors of 16
124816
4 × 4: the middle factor has no partner but itself.

Test 1,2,3,41, 2, 3, 4 in turn: 1×161 \times 16, 2×82 \times 8; 33 does not divide 1616 (16÷3=516 \div 3 = 5 remainder 11); 4×44 \times 4. The next number to test, 55, times itself is 2525, already bigger than 1616, so stop. The factors are 1,2,4,8,161, 2, 4, 8, 16: five of them, an odd number, because 44 pairs with itself.

Side note
Where it goes wrong

Mixing up the two words. Ask: is it smaller than the number (a factor, divides into it) or larger (a multiple, made from it)?

Try it
Try it: factors of 20

List all the factors of 2020 in pairs.

Show the answer

1×201 \times 20, 2×102 \times 10, 4×54 \times 5; 33 does not divide 2020. The next test, 55, already appeared in 4×54 \times 5, so the list is complete: 1,2,4,5,10,20\mathbf{1, 2, 4, 5, 10, 20}.

In Plain Terms & More Examples

Concept
In plain terms

A factor divides evenly with nothing left over; a multiple is what you get by counting up in steps of the number.

Example
The factors of 18

1×181 \times 18, 2×92 \times 9, 3×63 \times 6. So the factors are 1,2,3,6,9,181, 2, 3, 6, 9, 18.

Example
The first multiples of 4
04812162024+4+4+4+4+4+4
Counting in jumps of 4 lands on the multiples of 4.

Count up in fours: 4,8,12,16,20,…4, 8, 12, 16, 20, \dots

Side note
Where it goes wrong

Missing a pair. Listing the factors of 1818 as 1,2,9,181, 2, 9, 18 misses 3×63 \times 6. Test every whole number in order, 1,2,3,41, 2, 3, 4, and stop only when a number times itself is bigger than 1818 (5×5=255 \times 5 = 25).

Try it
Try it: multiples of 6

List the first five multiples of 66. Is 4242 a multiple of 66?

Show the answer

6×1,6×2,…,6×56 \times 1, 6 \times 2, \ldots, 6 \times 5: 6,12,18,24,30\mathbf{6, 12, 18, 24, 30}. 42=6×742 = 6 \times 7, so yes, 4242 is the seventh multiple of 66.

Prime or Composite?

Concept
Test small primes
1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950
The primes up to 50 are circled; every other number here (except 1) has a factor pair besides 1 × itself.

A number is prime if its only factors are 11 and itself. To test, try dividing by 2,3,5,72, 3, 5, 7: if none divides it and the number is less than 121121, it is prime.

Example
Is 9191 prime?

Not even; digits sum to 1010 so not divisible by 33; does not end in 00 or 55; 91÷7=1391 \div 7 = 13. So 91=7×1391 = 7\times13 is composite.

Example
Is 3737 prime?

Test the small primes in order. 22: 3737 is odd. 33: the digit sum is 3+7=103 + 7 = 10, not a multiple of 33. 55: 3737 does not end in 00 or 55. The next prime is 77, but 7×7=497 \times 7 = 49 is already bigger than 3737: any factor pair would need a partner smaller than 77, and those have all failed. So 3737 is prime.

Side note
Where it goes wrong

Two traps. First, 11 is not prime: it has only one factor, and a prime needs exactly two. Second, odd does not mean prime: 9=3×39 = 3 \times 3, 15=3×515 = 3 \times 5 and 21=3×721 = 3 \times 7 are all odd and all composite.

Try it
Try it: list the factors

List all the factors of 4848, and decide whether 5353 is prime.

Show the answer

Factor pairs of 4848: 1⋅48,2⋅24,3⋅16,4⋅12,6⋅81\cdot48, 2\cdot24, 3\cdot16, 4\cdot12, 6\cdot8 — factors 1,2,3,4,6,8,12,16,24,48\mathbf{1, 2, 3, 4, 6, 8, 12, 16, 24, 48}. 5353: not divisible by 2,3,52, 3, 5 or 77, so 53\mathbf{53} is prime.

Word Problems

Concept
Factor or multiple?

Splitting a group into equal parts with none left over asks for a factor. Things that repeat, like every 66 seconds or every 33rd day, and meet again ask for a multiple.

A factor fits exactly into a number, so it answers "how can I split this up?". A multiple is where you land by counting in equal steps, so it answers "when does this happen again?". Deciding which question the story asks is most of the work.

Example
Equal teams
123469121836
Each factor pair of 36 is a way to split the players: teams × players per team.

3636 players can form equal teams of 2,3,4,6,9,12,182, 3, 4, 6, 9, 12, 18 or 3636 — the factors of 3636 (other than 11).

Example
Party bags

2424 cookies and 1818 juice boxes go into identical bags with nothing left over. What is the greatest number of bags? The number of bags must be a factor of both numbers.

  • Factors of 2424: 1,2,3,4,6,8,12,241, 2, 3, 4, 6, 8, 12, 24.
  • Factors of 1818: 1,2,3,6,9,181, 2, 3, 6, 9, 18.

The greatest shared factor is 66. So 66 bags, each with 24÷6=424 \div 6 = 4 cookies and 18÷6=318 \div 6 = 3 juice boxes.

Side note
Where it goes wrong

Answering a factor question with a multiple. For the party bags, "7272 bags" (a common multiple of 2424 and 1818) cannot be right: you cannot make more bags than you have cookies. Splitting a group always gives a number no bigger than the group.

Try it
Try it: blinking lights

One light blinks every 66 seconds and another every 88 seconds. They blink together now. When do they next blink together?

Show the answer

The least common multiple of 66 and 88: multiples of 88 are 8,16,248, 16, 24; 2424 is the first that is also a multiple of 66. In 24\mathbf{24} seconds.

Common Multiples

Concept
Where two skip-counts meet

Skip-count by each number. Numbers on both lists are common multiples; the smallest one is the least common multiple (LCM).

A common multiple of 33 and 44 is a number you land on counting by 33s and counting by 44s. Picture two frogs starting together at 00, one jumping 33 at a time and one 44 at a time. The first stone they both touch is 1212. They meet again at 2424, 3636, and so on: every multiple of 1212.

Example
The least common multiple of 33 and 44
01224by 3sby 4s
Jumps of 3 above, jumps of 4 below: they first land together on 12.

Multiples of 33: 3,6,9,12,15,…3, 6, 9, 12, 15, \ldots Multiples of 44: 4,8,12,16,…4, 8, 12, 16, \ldots The first number on both lists is 1212.

Example
The least common multiple of 44 and 66

List multiples of the bigger number, 66: 6,12,18,246, 12, 18, 24. Check each against the 44s: 66 is not a multiple of 44; 12=4×312 = 4 \times 3 is. So the LCM is 1212, which is smaller than 4×6=244 \times 6 = 24.

Example
Hot dogs and buns

Hot dogs come in packs of 1010 and buns in packs of 88. What is the smallest number of each you can buy to have the same number of both? Multiples of 1010: 10,20,30,4010, 20, 30, 40. The first one that is also a multiple of 88 is 4040. Buy 40÷10=440 \div 10 = 4 packs of hot dogs and 40÷8=540 \div 8 = 5 packs of buns.

Side note
Where it goes wrong

Multiplying the two numbers and stopping. 4×6=244 \times 6 = 24 is always a common multiple of 44 and 66, but not always the least: 1212 comes first. Check the smaller multiples of the bigger number before settling.

Try it
Try it: 5 and 6

List the common multiples of 55 and 66 that are less than 7070. Which is the least?

Show the answer

Multiples of 66: 6,12,18,24,30,36,42,48,54,60,666, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66. The ones ending in 00 or 55 are also multiples of 55: 30\mathbf{30} and 60\mathbf{60}. The least common multiple is 30\mathbf{30}.

Why It Works

Factors come in pairs because a×b=na\times b = n means b=n÷ab = n \div a: every factor aa has a partner n÷an\div a. That is why you only need to test numbers up to the square root of nn to find all its factors — once aa passes n\sqrt n, its partner is smaller than n\sqrt n and was already found. For 3636, testing 11 through 66 finds the pairs (1,36),(2,18),(3,12),(4,9),(6,6)(1,36), (2,18), (3,12), (4,9), (6,6), and 66 pairs with itself because 3636 is a perfect square.

Tip
Reading a proof

Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.