Question

A number $a$ has 7 as its smallest prime factor, and another number $b$ has 3 as its smallest prime factor. What is the smallest prime factor of the number $a-b$?

• A. 1
• B. 2
• C. 3
• D. 5

Answer 1

The correct option is B

2

Since 7 is the smallest prime factor of $a$, it is clear that 2 is not a factor of $a$. Thus, $a$ is an odd number, i.e. $a=2n+1$ for any $n$.

Similarly, when 3 is the smallest prime factor of $b$, 2 cannot be a factor of $b$ and thus $b$ is also odd.i.e., $b=2m+1$ for any $m$.

Hence we have $a-b=(2n+1)-(2m+1)=2(n-m)=2k$ ( taking 2 common and assuming $k=n-m$.)

We can thus see that 2 is a factor of $a-b$, and hence 2 is the smallest prime factor of $a-b$.