If N=a^5b^4c where a, b, and c are three distinct prime numbers

Question

If  N=a^5b^4c  where a, b, and c are three distinct prime numbers, is N even?
1) The product of a, b, and c is even
2) The sum of a, b, and c is even

nick1816 · Answer

Question stem is whether atleast one among a,b and c is 2

Statement 1-

a*b*c = even

Hence atleast one of them is even or equals to 2

Sufficient

Statement 2-

a+b+c = even

Either one among them is even or all of them are even.

Sufficient

aloysiusgaul · Answer

1) The product of a, b, and c is even

Remember that an odd times an odd is odd, including by exponent. So 3*3 is odd and 9*3 is odd, so 3^3 is odd. The only way to get the product of these numbers to be even is to have an even sum among them. There are a few cases, but they all boil down to this: 
If a is odd, b is odd, and c is even, then a*b = odd. a*b*c then is the same as odd*even. We know that raising to an exponent doesn't change this rule (exponents are really just a form of multiplication anyway).   This is sufficient

2) The sum of a, b, and c is even

Remember that the exponents don't matter from, given what we discussed in statement 1. So, if a+b+c=even, let's do the same thing as before. Let's say a and b are odd, so a+b = even. What if a and b are even but c is odd? Then even + odd = odd, so we know this can't be the case. Either all of the numbers are even or only one of the numbers is even. If all of them are even, obviously the product is even. But recall that these are positive prime numbers, and there is only one even prime number anyway, so all being even isn't possible . If one is even and the other two are odd, then we are left with the same example as statement 1: a*b = odd, so odd*even = even.   This is sufficient

If N=a^5b^4c where a, b, and c are three distinct prime numbers

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

If N=a^5b^4c where a, b, and c are three distinct prime numbers

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards