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# If n is a positive integer, is 169 a factor of n?

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If n is a positive integer, is 169 a factor of n?  [#permalink]

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24 Jul 2019, 07:00
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85% (hard)

Question Stats:

65% (02:18) correct 35% (02:01) wrong based on 250 sessions

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If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
(2) 1,352 is the least common multiple of 104 and n

 This question was provided by Math Revolution for the Game of Timers Competition

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Re: If n is a positive integer, is 169 a factor of n?  [#permalink]

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24 Jul 2019, 07:18
2
If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
260 = 52*5
Possible values of
n = 52 —> 169 is not a divisor of n
n = 52*13 —> 169 is a divisor of n

Insufficient

(2) 1,352 is the least common multiple of 104 and n

1352 = 169*8 = $$13^2$$*8
104 = 13*8
Possible values of
n = $$13^2$$ —> 169 is a factor of n
n = $$13^2$$*2 —> 169 is a factor of n
n = $$13^2$$*4 —> 169 is a factor of n
n = $$13^2$$*8 —> 169 is a factor of n

Sufficient

IMO Option B

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Re: If n is a positive integer, is 169 a factor of n?  [#permalink]

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24 Jul 2019, 07:23
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(1) 52 is the greatest common divisor of 260 and n
not sufficient

(2) 1,352 is the least common multiple of 104 and n
divide number by 104 and 169 you can tell the number is divisible by 169
sufficient
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Re: If n is a positive integer, is 169 a factor of n?  [#permalink]

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24 Jul 2019, 08:05
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If n is a positive integer, is 169 a factor of n?
Given: n > 0 & integer
n = 169*m ? where m is an integer from 1 to any m
=> n = 13^2*m?

(1) 52 is the greatest common divisor of 260 and n --> insufficient: 260 = 52*5, so GCD of 52*5 and n = 52, so n = 52*p=13*4*p, but p can or can't be multiple of 13
(2) 1,352 is the least common multiple of 104 and n --> sufficient: 104 = 13*2^3, so LCM of 13*2^3 & n = 1,352 = 13^2*2^3, so n must be multiple of 13^2

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Re: If n is a positive integer, is 169 a factor of n?  [#permalink]

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24 Jul 2019, 08:29
1
If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
52=2*2*13
n=$$2^2*13k$$
But 169=$$13^2$$
and n is not necessarily a multiple of 169
or 169 is not necessarily a factor of n
NOT SUFFICIENT

(2) 1,352 is the least common multiple of 104 and n
1352=13*104 = $$2^3*13^2$$
104=2*2*2*13 = $$2^3*13$$
Since 1352 is least common multiple of 104 and n
=>$$13^2$$ is a factor of n
=> 169 is a factor of n
SUFFICIENT

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Re: If n is a positive integer, is 169 a factor of n?  [#permalink]

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25 Jul 2019, 04:26
1
If n is a positive integer, is 169 a factor of n?

(1) 52 is the greatest common divisor of 260 and n
(2) 1,352 is the least common multiple of 104 and n

Question Stem Analysis:

According to the stem, we must find out whether n/169 , i.e $$\frac{n}{13^2}$$ is an integer.

Statement One Alone:

Prime factorization of 52 is 13 X $$2^2$$
Prime factorization of 260 is 13 X 5 X $$2^2$$

The greatest common divisor (gcd), also known as the greatest common factor (gcf), or highest common factor (hcf), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

For GCD, we pick the lowest power of common factors. we know that n consist a 13, but n can or cannot contain more than $$13^2$$, because gcf of 260 and n is limited to 13 which we have in 260.

Hence this statement alone is not sufficient.

Statement Two Alone:

Prime factorization of 104 is 13 X$$2^3$$
Prime factorization of 1352 is 13 X 13 X $$2^3$$

The lowest common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple both of a and of b.

For LCM, we pick the highest power of common factors. we know that 104 just has $$13^1$$, and that 1352 has $$13^2$$ which definitively must have come from n. Hence we can say that n has$$13^2$$ and that will make it divisible by 169.

Statement two alone is sufficient.

Hence the answer must be B
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Re: If n is a positive integer, is 169 a factor of n?   [#permalink] 25 Jul 2019, 04:26
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