Primes, primes and?

Number Theory Level 3

Let $m$ and $n$ be positive integers, each of which has 7 as its least prime factor.

What is the least prime factor of $m + n?$

2 3 5 7 Can't be determined

4 solutions

Santu Paul
Dec 26, 2018

Since least prime factor of m and n is 7.

Therefore, neither of them have two as a factor.

Therefore, m and n are odd.

We know, $odd + odd = even$

$\Rightarrow (m + n)$ is even.

The required answer is $\boxed{2}$

Therefore, neither of them have two as a factor.

This is where you run astray. One of them could have two as a factor, so long as the other one doesn't.

Jordan Cahn - 2 years, 5 months ago

'both m and n' is mentioned in the question.

SANTU PAUL - 2 years, 5 months ago

Mr. India
Jan 22, 2019

Since least prime factor of m and n is 7, they don't have 2 as a factor and hence, are odd.

So let m = 2a + 1 and n = 2b + 1

m + n = 2a + 2b + 2

       = 2 ( a+ b+ 1 )

Hence, m + n has 2 as it's least prime factor.

Note : 2a + 1 n 2b + 1 are general forms of odd numbers ( a and b are whole numbers )

Deepak Maurya
Dec 27, 2018

Since both nos. Have 7 as least factor so they are odd. Odd+ Odd = even. So answer is 2.

Edwin Gray
Dec 26, 2018

Let m = n. Then the lowest prime factor of m + n is 2. Ed Gray