GCF and LCM — Challenge

Prime factorisation answers both questions at once: shared factors give the GCF, and everything needed gives the LCM.

What these problems cover

These questions use 5 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.

Problem 1

What is the greatest common factor of 12 and 18?

  1. 2
  2. 3
  3. 6
  4. 36
  5. 216
Worked answer

6

List the factors: 12 has 1,2,3,4,6,12 and 18 has 1,2,3,6,9,18.

The common factors are 1,2,3,6, and the greatest of them is 6.

36 is the least common multiple, which is a different question.

Problem 2

What is the least common multiple of 6 and 8?

  1. 2
  2. 12
  3. 24
  4. 48
  5. 14
Worked answer

24

List multiples of 8: 8, 16, 24, … and check which is also a multiple of 6.

24 is the first one, since 24 = 6· 4 = 8· 3.

48 is a common multiple but not the least one.

Problem 3

What is the greatest two-digit number divisible by both 2 and 3?

  1. 90
  2. 92
  3. 94
  4. 96
  5. 98
Worked answer

96

A number divisible by both 2 and 3 is divisible by their least common multiple, which is 6.

The largest two-digit number is 99, and the largest multiple of 6 at or below it is 96, since 96 = 6· 16.

98 is even but not divisible by 3, because its digits sum to 17.

Problem 4

What is the greatest common factor of 20 and 30?

  1. 2
  2. 5
  3. 10
  4. 60
  5. 600
Worked answer

10

Both numbers are divisible by 10, since 20 = 10· 2 and 30 = 10· 3.

No larger number divides both, because 2 and 3 share no factor.

So the greatest common factor is 10.

Problem 5

What is the least common multiple of 4 and 10?

  1. 2
  2. 14
  3. 20
  4. 40
  5. 80
Worked answer

20

Multiples of 10 are 10, 20, 30, …; the first that 4 also divides is 20.

Check: 20 = 4· 5 = 10· 2.

Multiplying 4 by 10 gives 40, a common multiple, but not the least, because 4 and 10 share the factor 2.

Problem 6

Two bells ring together at noon. One rings every 12 minutes and the other every 18 minutes. After how many minutes do they next ring together?

  1. 6
  2. 30
  3. 36
  4. 72
  5. 216
Worked answer

36

They ring together at times that are multiples of both 12 and 18, so the answer is the least common multiple.

Multiples of 18 are 18, 36, … and 36 is also 12· 3.

So they next ring together after 36 minutes. Adding the two intervals to get 30 is a common error.

Problem 7

What is the greatest common factor of 7 and 13?

  1. 1
  2. 7
  3. 13
  4. 20
  5. 91
Worked answer

1

Both 7 and 13 are prime, so each has only itself and 1 as factors.

They are different primes, so the only factor they share is 1.

Numbers whose greatest common factor is 1 are called relatively prime.

Problem 8

What is the least common multiple of 9 and 12?

  1. 3
  2. 21
  3. 36
  4. 72
  5. 108
Worked answer

36

Write each as a product of primes: 9 = 32 and 12 = 22· 3.

The least common multiple takes the highest power of each prime: 22· 32 = 36.

108 is 9· 12, a common multiple but not the least, because they share a factor of 3.

Problem 9

What is the smallest whole number greater than 1 that leaves a remainder of 1 when divided by each of 2, 3, 4, 5 and 6?

  1. 31
  2. 41
  3. 51
  4. 61
  5. 121
Worked answer

61

If the number leaves remainder 1 in every case, then one less than it is divisible by all of 2,3,4,5,6.

The least common multiple of those is 60, so the number one less is 60 and the number itself is 61.

Check: 61 = 60+1, and 60 is divisible by each of the five divisors.

Problem 10

What is the greatest common factor of 16 and 24?

  1. 2
  2. 4
  3. 8
  4. 48
  5. 384
Worked answer

8

Write each as a product of primes: 16 = 24 and 24 = 23· 3.

The greatest common factor takes the lowest power of each shared prime, which is 23 = 8.

Check: 16 = 8· 2 and 24 = 8· 3.

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