HCF and LCM: 17 MCQs with answers
All 17 multiple-choice questions from this kit, free and with no sign-up.
Warm-up · 5 questions
Q1. What is the HCF of $12$ and $18$?
BeginnerShow the answer
A · $6$
The largest number dividing both $12$ and $18$ is $6$ (since $12 = 6 \times 2$ and $18 = 6 \times 3$).
Q2. What is the LCM of $12$ and $18$?
BeginnerShow the answer
D · $36$
The smallest number that both $12$ and $18$ divide into is $36$.
Q3. Which statement correctly describes the LCM of two numbers?
BeginnerShow the answer
B · The smallest number that both divide into exactly
The LCM is the smallest number that is a multiple of both given numbers.
Q4. Two bells ring every $12$ minutes and every $18$ minutes. They ring together now. After how many minutes will they next ring together?
BeginnerShow the answer
B · $36$ minutes
The next simultaneous ring is the LCM of $12$ and $18$, which is $36$ minutes.
Q5. For any two positive integers $a$ and $b$, $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.
BeginnerShow the answer
A · True
For any two positive integers $a$ and $b$, $\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$.
Building up · 7 questions
Q6. For $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$, what is the HCF?
IntermediateShow the answer
B · $2^2 \times 3 = 12$
HCF takes shared primes at their lowest powers: $2^2 \times 3 = 12$.
Q7. For $24 = 2^3 \times 3$ and $36 = 2^2 \times 3^2$, what is the LCM?
IntermediateShow the answer
D · $72$
LCM takes all primes at their highest powers: $2^3 \times 3^2 = 72$.
Q8. A student finds the HCF of $12$ and $18$ and gets $36$. Which sanity check proves this wrong?
IntermediateShow the answer
C · The HCF can never exceed the smaller number
The HCF is a divisor, so it cannot be larger than the smaller number. $36$ cannot divide $12$.
Q9. If $\text{HCF}(a,b) = 4$ and $a \times b = 96$, what is $\text{LCM}(a,b)$?
IntermediateShow the answer
A · $24$
Using $\text{HCF} \times \text{LCM} = a \times b$: $4 \times \text{LCM} = 96$, so $\text{LCM} = 24$.
Q10. Which pair of numbers is coprime?
IntermediateShow the answer
B · $8$ and $15$
$8 = 2^3$ and $15 = 3 \times 5$ share no prime factor, so their HCF is $1$ — they are coprime.
Q11. To simplify $\frac{18}{24}$ in one step, which value should you divide numerator and denominator by?
IntermediateShow the answer
B · The HCF of $18$ and $24$
Dividing by the HCF ($6$) gives $\frac{3}{4}$ directly in one step.
Q12. A carpenter has boards of length $48$ cm and $18$ cm and wants the largest identical planks that fit both exactly. Which tool does he need?
IntermediateShow the answer
B · The HCF of $48$ and $18$
Cutting the largest identical pieces is a splitting problem, so the HCF ($6$ cm) is the answer.
Topper level · 5 questions
Q13. For $4$, $6$ and $8$, the HCF is $2$ and the LCM is $24$. Which statement is true?
AdvancedShow the answer
B · $\text{HCF} \times \text{LCM} \ne 4 \times 6 \times 8$ because the product rule needs exactly two numbers
$2 \times 24 = 48$ but $4 \times 6 \times 8 = 192$. The product rule applies only to two numbers.
Q14. Two gears have $24$ and $36$ teeth. A marked tooth on each is aligned. After how many tooth-passes will the marked teeth next align?
AdvancedShow the answer
B · $72$ tooth-passes
Alignment recurs at the LCM of $24$ and $36$, which is $72$.
Q15. Applying the Euclidean algorithm to $48$ and $18$: $48 = 18 \times 2 + 12$, then $18 = 12 \times 1 + 6$, then $12 = 6 \times 2 + 0$. What is the HCF?
AdvancedShow the answer
D · $6$
The last non-zero remainder is $6$, so $\text{HCF}(48, 18) = 6$.
Q16. For two positive integers, which statement best explains why a large HCF forces a small LCM?
AdvancedShow the answer
B · Because their product is fixed at $a \times b$, so one must shrink as the other grows
The product rule $\text{HCF} \times \text{LCM} = a \times b$ fixes the product, so the two values trade off.
Q17. Given two integers, $a = 24$ and $b = 36$, calculate the HCF and LCM. Then, if the product of the two integers is $a \times b$, verify the product rule. What is the value of $\text{LCM}(a,b)$?
AdvancedShow the answer
A · 72
The HCF of $24$ and $36$ is $12$, and the LCM is calculated as $\text{LCM}(a,b) = \frac{a \times b}{\text{HCF}(a,b)} = \frac{24 \times 36}{12} = 72$. Thus, the correct value of $\text{LCM}(24,36)$ is $72$.
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