About Problems on H.C.F and L.C.M
The highest common factor is the largest number that divides both given numbers. The lowest common multiple is the smallest number that both of them divide. They are tied together by one relation that settles most questions in the topic: the product of two numbers equals their H.C.F multiplied by their L.C.M.
What you need to understand
- The H.C.F of two numbers is the largest number dividing both of them exactly.
- The L.C.M of two numbers is the smallest number divisible by both of them.
- For any two numbers, product = H.C.F x L.C.M. This is the most used fact in the topic.
- The H.C.F is never larger than the smaller of the two numbers, and the L.C.M is never smaller than the larger.
- Two numbers whose only common factor is 1 are called coprime, and their L.C.M is simply their product.
- Bells tolling together, lights flashing together and wheels meshing are all L.C.M questions in disguise, because a repeating interval is a common multiple.
Formulas to remember
How to work through these questions
- For a question asking for the greatest number that divides or measures something, look for the H.C.F.
- For a question asking for the smallest number divisible by several numbers, or for when repeating events next coincide, look for the L.C.M.
- When the product and one of the two statistics is given, use product = H.C.F x L.C.M to find the other straight away.
- To find an L.C.M by factors, break both numbers into prime factors and take the highest power of every prime that appears.
- Check that the H.C.F divides both numbers and the L.C.M is a multiple of both, which catches most slips.
Mistakes that cost marks
- Multiplying the two numbers and calling it the L.C.M when they are not coprime.
- Giving the H.C.F when the question asks for the L.C.M, or the other way round.
- Forgetting to add the remainder back when a question asks for the least number leaving a stated remainder.
- Assuming the L.C.M is always the product of the two numbers.
- Taking the L.C.M when the question asks for the greatest common divisor.
Worked example
The product of two numbers is 216 and their L.C.M is 36. Find their H.C.F.
- For any two numbers, product = H.C.F x L.C.M.
- So H.C.F = product / L.C.M = 216 / 36.
- Since 36 x 6 = 216, the H.C.F is 6.
- Check: the numbers could be 12 and 18, whose H.C.F is 6 and whose L.C.M is 36. Both conditions hold.
Answer: 6
Practice questions with answers
A few Problems on H.C.F and L.C.M questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
What is the least common multiple of 28 and 68?
Answer: Option B — with explanation
28 = 4 x 7 and 68 = 4 x 17.
L.C.M = (product of the numbers) / H.C.F = (28 x 68) / 4 = 476.
The H.C.F of the two numbers is 4.
Common mistakes
- multiplied the two numbers, which gives the L.C.M only when they are coprime: 1904
- gave the larger of the two numbers: 68
- added the two numbers: 96
Question 2
What is the highest common factor of 74 and 51?
Answer: Option C — with explanation
74 and 51 share no common factor other than 1, so the highest common factor is 1.
The L.C.M of the two numbers is 3774.
Common mistakes
- added the two numbers: 125
- gave the L.C.M instead of the H.C.F: 3774
- gave the smaller of the two numbers: 51
Question 3
Two numbers have a product of 5376 and a least common multiple of 1344. What is their highest common factor?
Answer: Option C — with explanation
For any two numbers, product = H.C.F x L.C.M.
So H.C.F = 5376 / 1344 = 4.
Check: the numbers are 84 and 64, whose H.C.F is 4 and L.C.M is 1344.
Common mistakes
- added the two numbers: 148
- halved the H.C.F for no reason: 2
- gave the product of the two numbers: 5376
Question 4
Find the L.C.M of 49 and 85.
Answer: Option C — with explanation
49 and 85 share no common factor other than 1, so their L.C.M is simply their product, 49 x 85 = 4165.
The H.C.F of the two numbers is 1.
Common mistakes
- gave the larger of the two numbers: 85
- gave the H.C.F instead of the L.C.M: 1
- added the two numbers: 134
Question 5
Find the greatest number that divides both 23 and 69 exactly.
Answer: Option D — with explanation
23 = 23 x 1 and 69 = 23 x 3.
The largest number dividing both is therefore 23.
The L.C.M of the two numbers is 69.
Common mistakes
- subtracted the two numbers: 46
- gave the L.C.M instead of the H.C.F: 69
- added the two numbers: 92
Frequently asked questions
What is the relation between the H.C.F and the L.C.M of two numbers?
Their product equals the product of the two numbers. This is the most useful fact in the topic, and it holds for two numbers only, not for three.
When should I look for the H.C.F and when for the L.C.M?
Look for the H.C.F when the question asks for the greatest number that divides something, or for the largest tile or length that fits evenly. Look for the L.C.M when it asks for the smallest number divisible by several numbers, or for when repeating events next coincide.
Are the H.C.F and the L.C.M ever equal?
Only when the two numbers are identical. In every other case the H.C.F is smaller than both numbers and the L.C.M is larger than both.
What if the two numbers are coprime?
Then their H.C.F is 1 and their L.C.M is their product. Coprime pairs such as 8 and 15 need no factorisation at all.
