(D/S) Root N = interger?

Question

Dear Math Masters! Please help me out...

Q) If N= R^a*S^b, R and S are primes, a and b are positive intergers. Is root N is an integer?

1) a+b= even
2) ab= even

Thank you for your answer and explanations in advance!

walker · Answer

C.

root(N) is integer if a and b both are even

1. even,even and odd,odd - appropriate. insufficient
2. even,even and odd,even - appropriate. insufficient

1&2. only even,even - sufficient.

GMATBLACKBELT · Answer

1) 2^3*3^3 = sqrt216. Not an integer. 2^1*2^1--> sqrt4 =2

2) Same thing 2^1*2^1=4 --> sqrt =2

2^1*2^2 --> 8--> sqrt 8 dsnt equal an integer.

Together a and b can only be even.

So C.

pretttyune · Answer

Relived now! OA is C right! THANKS!!

pmenon · Answer

ok, i got as far as identifying that both a and b had to be even , but how does that help us determine whether root(N) is an integer ?

walker · Answer

for any integer: n=a^i*b^m*...*c^n,
where a,b,...,c - prime numbers;
i,m,...,n - powers of prime numbers.

n^2=a^2i*b^2m*...*c^2n, so all powers is even.

Therefore, a square root of integer N will be also an integer if all powers of N prime numbers are even.

pmenon · Answer

thanks, walker. is that a rule, because i wasnt aware of this property previously.

(D/S) Root N = interger?

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