carcass wrote:

If m and n are positive integers, is m + n divisible by 4 ?

(1) m and n are each divisible by 2.

(2) Neither m nor n is divisible by 4.

Statement 1- Let's say m = (2*a*b*...), which a,b,... are prime numbers.

- Let's say n = (2*x*y*...), which x,y,... are prime numbers.

- \(\frac{(m+n)}{4}\) = \(\frac{(2*a*b*...) + (2*x*y*...)}{4}\) = \(\frac{2 ( (a*b*...) + (x*y*...))}{4}\)

- Here we can see that a,b,x,y can be anything prime numbers.

- #1 If (a*b) + (x*y) = EVEN, YES, \(\frac{m+n}{4}\) divisible by 4.

- #2 If (a*b) + (x*y) = ODD, NO, \(\frac{m+n}{4}\) cannot divisible by 4.

- Hence, INSUFFICIENT.

Statement 2- Each of m and n has only one maximum one factor of 2 so it cannot divisible by 4.

- Try to plug simple number : m = 3 and n = 5, divisible by 4, YES.

- Try to plug another random number : m = 5 and n = 9, NOT divisible by 4, NO.

- Hence, INSUFFICIENT.

Both statement- Go back to this equation :

- \(\frac{(m+n)}{4}\) = \(\frac{(2*a*b*...) + (2*x*y*...)}{4}\) = \(\frac{2 ( (a*b*...) + (x*y*...))}{4}\)

- If m & n divisible by 2 but not by 4, so a,b,x,y MUST BE ODD.

- (ODD*ODD) + (ODD*ODD) = ODD + ODD = EVEN.

- 2 * ANY even number MUST BE DIVISIBLE BY 4.

- Hence, SUFFICIENT.

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