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.
Which of these is a common multiple of $3$ and $5$?
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.
Which pair of numbers has exactly two common factors?
Correct answer: C - $10$ 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 $1$.
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.
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.
Two numbers have a highest common factor of $1$. What can be said about them?
Correct answer: D - 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.
Which of these numbers has exactly three factors?
Correct answer: D - $25$
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 $1, 3, 9, 27$.
What is the highest common factor of $8$ and $12$?
Correct answer: B - $4$
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.
What is the largest factor of any whole number greater than $1$?
Correct answer: D - 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.
How many common multiples do two whole numbers have?
Correct answer: B - 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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