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If positive integer N=A–B, where A is a positive integer a

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If positive integer N=A–B, where A is a positive integer a

by BTGmoderatorDC » Thu Feb 21, 2019 6:01 pm

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If positive integer N=A-B, where A is a positive integer and B is a prime number, is N odd?

(1) B and X are the only prime factors of A, and B-X=4

(2) A is divisible by 9 numbers in total, one of which is B^2

OA A

Source: e-GMAT
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Source: — Data Sufficiency |
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edited

by deloitte247 » Sun Feb 24, 2019 9:16 am

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Statement 1
B and X are the only prime factor of A, and B - X = 4; this mean B and X are odd numbers and one can conclude that
$$A=B^n\cdot X^k$$
B is prime, X is a prime factor so X > 1 , B and X are two different terms so X = B - 4 , so X = odd prime and so is B
Therefore,
$$A=Odd\cdot Odd=Odd^n\left(where\ Odd^n\ regardless\ of\ the value\ of\ n\ =odd\right)$$
Hence,
$$N=A\left(odd\right)-B\left(odd\right)=even$$
$$So\ N\ne Odd,\ statement\ 1\ is\ SUFFICIENT.$$

Statement 2
A is divisible by 9 numbers in total
$$One\ of\ which\ is\ B^2$$
This means factors of A includes the following :-
$$1,\ B,\ B^2,\ A------$$
Assuming that
$$A=B^2\cdot k\ \left(where\ k=k+1=1,2,3,4\right)$$
$$N=B^2\cdot k-B$$
$$=B\left(Bk-1\right)\ where\ Bk-1=Bk-odd$$
If k= odd then N = odd - odd = even
If k = even then N = even - odd = odd
Statement 2 is INSUFFICIENT.
Statement 1 alone is SUFFICIENT.

$$answer\ is\ Option\ A$$
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