What smallest positive integer value, when multiplied with 350, will r

Question

What smallest positive integer value, when multiplied with 350, will result in a number with odd number of factors? A) 2 B) 7 C) 10 D) 14 E) 35 Source: https://www.GMATinsight.com

ArunSharma12 · Answer

perfect square have odd number of factors.
350 = 7*5*5*2
350 requires a 2 and 7 prime factor to be a perfect square.
 Ans: 14

Archit3110 · Answer

odd number of factors are of perfect square integers only ; 
350 ; 2*5^2*7 ; 
350 is to be multiplied by 2*7 ; 14
IMO D ; 
 GMATinsight  sir is oa correct?

saumyagupta1602 · Answer

The correct option is D 14 however the answer choice next to the timer is showing option c 10 as the correct answer, which is incorrect.

fireagablast · Answer

I'm confused. 
350 = 2*5*5*7 --> is this 3 factors or 4?

Either way
350*2 = 700 = 2*2*5*5*7 --> this is 3 factors or 5?

Why is 2 not the answer if the number of factors is odd?

Edit: I understand now
the prime factorization of the number does not represent all the factors or relate to even/odd number of total factors
700 --> 2*350 --> 2*175 --> 5*35 --> 7*5 --> factors are 1, 2, 5, 7, 35, 175, 350, 700 which is an even number of factors

As people have stated above only perfect squares have odd total number of factors.
To have a perfect square, you need an equal distrubution of each prime factors
350 = 2*5*5*7 --> need to equalize the 2's and 7's to match the 5's
14*350 = 2*7*350= 2*2*5*5*7*7 --> this will be a perfect square and therefore have an odd number of total factors.

Proof:
16 = 1, 2, 4, 8, 16 (odd)
9 = 3 (odd)
26 = 1, 2, 3, 6, 36 (odd)
14 = 1, 2, 7, 14 (even)
15 = 1, 3, 5, 15 (even)

LeoN88 · Answer

Any perfect square will have odd number of factors:-

Check:-

24: 
1*24
2*12
3*8
4*6

13: 
13*1

49: 
1*49
7*7

Now we have: 350 = 2 * 5^2 * 7
How to make it an immediate perfect square? Multiply with one 2 and one 7-
2^2 * 5^2 * 7^2

Bunuel  could you check OA?

cjk52793 · Answer

2^1 • 5^2 • 7^1 is the prime factorization of 350.

Total Factors=(1+1)(2+1)(1+1)=12.

For the number of factors to be odd, (n+1)(n+1)..., where n is the exponent of the prime factors, must be odd, so no odd exponent of a prime factor will result in an odd number of factors.

D) 14=2•7
(14)(350) = (2^2)(5^2)(7^2) = (3)(3)(3) factors = 27 factors.

IMO: D

Posted from my mobile device

GMATinsight · Answer

Looks like I pressed the wrong button while marking OA.

saumyagupta1602   Archit3110   LeoN88

Have made the correction. Sorry for the mistake...

GMATinsight · Answer

Concept 1: Perfect Square (Squares of Integers) have ODD number of factors   The derivation is as mentioned in Video below

https://www.youtube.com/watch?v=DBkIP8Fg20s

i.e. We need to multiply a smallest positive integer to obtain a perfect square result

CONCEPT 2: Perfect squares have EVEN exponents of all its distinct prime factors

350 = 2*5^2*7

While 5 has even exponent, 2 and 7 have power one each (odd exponents)

i.e. to make the exponents of 2 and 7 even, we need to multiply one 2 and one 7

and 2*7 = 14

i.e. we need to multiply 14 with 350 to obtain a perfect square result which will have ODD number of factors

Answer: Option D

GMATinsight · Answer

This problem is a combination of two of my questions posted on GC in past the links are mentioned below

Problem 1: https://gmatclub.com/forum/for-what-sma ... fl=similar

Problem 2: https://gmatclub.com/forum/how-many-int ... l#p2470658

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What smallest positive integer value, when multiplied with 350, will r

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What smallest positive integer value, when multiplied with 350, will r

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