If m is a positive integer, is √m a prime number?

Question

I. √m leaves a remainder of 1 when divided by 6
II. m has 3 factors

IanStewart · Answer

Using Statement 1, √m could be 7, and could be prime, or √m could be 25, and could be non-prime.

The only numbers with precisely 3 divisors are the squares of primes: if p is prime, the number p^2 has the divisors 1, p and p^2 (any number with two distinct prime divisors p and q will always have at least the four divisors 1, p, q and pq, so our number here can only have one prime factor p, and if it has exactly 3 divisors, it must be equal to p^2). So Statement 2 is sufficient.

bond001 · Answer

IanStewart,buneul
i found that answer should be D
I was not able to prove statement 1 false i only got three numbers that are prime 
25,49,121 and square root of them is prime numbers 
statement 2 is sufficient 
because a perfect square prime has three factors 
for me the answer is D, please help

IanStewart · Answer

First, we need it to be true that √m gives us a remainder of 1 when we divide by 6. m cannot equal 25 or 121 here, because then √m would be 5 or 11, and those numbers give us a remainder of 5 when we divide by 6.

If the remainder is 1 when we divide √m by 6, then √m could be any number in this list:

1, 7, 13, 19, 25, 31, 37, 43, 49, ...

Since only some of these numbers are prime, Statement 1 is not sufficient (it might be that m = 1, or m = 625, and then √m is not prime).

bond001 · Answer

Thanks IanStewart got it

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If m is a positive integer, is √m a prime number?

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