Question

A number $a$ has 3 as its smallest prime factor, and another number $b$ has 5 as its smallest prime factor. What is the smallest prime factor of the number $a+b$?

• A. 10
• B. 5
• C. 3
• D. 2

Answer 1

The correct option is D

2

Since 3 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 some $n$.
Similarly, when 5 is the smallest prime factor of $b$, 2 cannot be a factor of $b$ and thus $b$ too is odd.i.e., $b=2m+1$ for some $m$.
Thus, $a+b=(2n+1)+(2m+1)=2(n+m+1)=2k\left(say\right)$ where $k=n+m+1$.
We can thus see that 2 is a factor of $a+b$, and clearly 2 is the smallest prime factor of $a+b$.