common-multiples-and-common-factors MCQs for UPSC Prelims

32 practice questions on common-multiples-and-common-factors from the Prime Time section of the UPSC Prelims syllabus. 32 come with a written explanation. Try the sample set below - the answer stays hidden until you ask for it.

12 Easy 14 Medium 6 Hard

Sample questions

Q1
easy
Which of these is a common multiple of $3$ and $5$?
  1. A $25$
  2. B $35$
  3. C $45$
  4. D $21$
Show answer and explanation

Correct answer: C - $45$

Since $45 = 3 \times 15$ and $45 = 5 \times 9$, it is a multiple of both. The number $25$ is a multiple of $5$ only, $21$ of $3$ only, and $35$ of $5$ only.
Q2
hard
Which pair of numbers has exactly two common factors?
  1. A $12$ and $18$
  2. B $8$ and $12$
  3. C $7$ and $11$
  4. D $10$ and $15$
Show answer and explanation

Correct answer: D - $10$ and $15$

The common factors of $10$ and $15$ are $1$ and $5$, exactly two. The pair $8$ and $12$ shares $1, 2, 4$, the pair $12$ and $18$ shares $1, 2, 3, 6$, and $7$ and $11$ share only $1$.
Q3
hard
Assertion (A): For any two whole numbers, their highest common factor is never greater than either number. Reason (R): A factor of a number can never exceed that number.
  1. A Both A and R are true, and R is the correct explanation of A
  2. B Both A and R are true, but R is not the correct explanation of A
  3. C A is true, but R is false
  4. D A is false, but R is true
Show answer and explanation

Correct answer: A - Both A and R are true, and R is the correct explanation of A

Both statements are true, and R explains A directly: since the highest common factor divides each number, it cannot exceed either of them. Option (b) fails because that is exactly the reason, and neither statement is false.
Q4
hard
Two numbers have a highest common factor of $1$. What can be said about them?
  1. A One must be $1$
  2. B They share no factor other than $1$
  3. C They must be consecutive
  4. D Both must be prime
Show answer and explanation

Correct answer: B - They share no factor other than $1$

A highest common factor of $1$ means no larger number divides both, which is the definition of co-prime. The numbers need not be prime, as $8$ and $9$ show, neither has to equal $1$, and they need not be consecutive, as $8$ and $15$ show.
Q5
hard
Which of these numbers has exactly three factors?
  1. A $15$
  2. B $24$
  3. C $25$
  4. D $27$
Show answer and explanation

Correct answer: C - $25$

A number has exactly three factors only when it is the square of a prime, and $25$ gives $1, 5$ and $25$. The number $24$ has eight factors, $15$ has four, and $27$ has four, namely $1, 3, 9, 27$.
Q6
easy
What is the highest common factor of $8$ and $12$?
  1. A $24$
  2. B $2$
  3. C $4$
  4. D $8$
Show answer and explanation

Correct answer: C - $4$

The common factors of $8$ and $12$ are $1, 2$ and $4$, so the highest is $4$. Answering $2$ picks a smaller common factor, $8$ does not divide $12$, and $24$ is their least common multiple.
Q7
medium
What is the largest factor of any whole number greater than $1$?
  1. A The number itself
  2. B The number minus $1$
  3. C $1$
  4. D Half the number
Show answer and explanation

Correct answer: A - The number itself

Every number divides itself exactly once, and nothing larger can divide it, so the number itself is its greatest factor. One is the smallest factor, half the number is a factor only when the number is even, and the number minus one usually is not a factor at all.
Q8
easy
How many common multiples do two whole numbers have?
  1. A Exactly two
  2. B None, unless one divides the other
  3. C Exactly one
  4. D Infinitely many
Show answer and explanation

Correct answer: D - Infinitely many

Once you have one common multiple, doubling and tripling it gives more without end, so there are infinitely many. Exactly one would be the lowest common multiple alone, and every pair of whole numbers has at least their product in common.

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