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12 May 2020, 09:15
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If the integers a and n are greater than 1 and the product of the first 8 positive integers is a multiple of a^n, what is the value of a ?
(1) a^n = 64
(2) n = 6
*********
from stem: 1*2*3*...*8= (a^n).k
(2^7)*(3^2)*(5)*(7) = (a^n).k
1) a^n = 64, prime factorizing 64 = 2^6
(2^7)*(3^2)*(5)*(7) = (a^n).k
-> (2^7)*(3^2)*(5)*(7) = 64 .k
-> (2^7)*(3^2)*(5)*(7) = (2^6).k
Only possible if a=2
2) n=6
(2^7)*(3^2)*(5)*(7) = (a^6).k
Only possible when a=2
Thus answer should be D..
But, the solution says Answer is B.
Could u please explain?
(1) a^n = 64
(2) n = 6
*********
from stem: 1*2*3*...*8= (a^n).k
(2^7)*(3^2)*(5)*(7) = (a^n).k
1) a^n = 64, prime factorizing 64 = 2^6
(2^7)*(3^2)*(5)*(7) = (a^n).k
-> (2^7)*(3^2)*(5)*(7) = 64 .k
-> (2^7)*(3^2)*(5)*(7) = (2^6).k
Only possible if a=2
2) n=6
(2^7)*(3^2)*(5)*(7) = (a^6).k
Only possible when a=2
Thus answer should be D..
But, the solution says Answer is B.
Could u please explain?
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12 May 2020, 15:08
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This question is discussed here: https://gmatclub.com/forum/if-the-integ ... 32382.html
You can post your doubts on the above-mentioned thread.
Kindly go through the rules: https://gmatclub.com/forum/rules-for-po ... 33935.html
Do search the forum before posting
You can post your doubts on the above-mentioned thread.
Kindly go through the rules: https://gmatclub.com/forum/rules-for-po ... 33935.html
Do search the forum before posting
12 May 2020, 23:44
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Rishab130612
Hello Rishab,
The product of the first 8 integers is 8!. The question says that 8! is a multiple of \(a^n\) where a and n are integers greater than 1.
This means, a can be 2 or 3 or 4 and so on till 8, right?
From statement I alone, \(a^n\) = 64. This is where you have restricted yourself to\( 2^6\). Why can’t 64 be \(4^3\) or why not \(8^2\)?
It also appears like you have subconsciously used the data given in statement II, otherwise, you would have thought more openly about 64.
Now you understand why statement I alone is insufficient, right?
From statement II alone, n = 6. This can happen only with \(2^6\). You ask why?
Because the highest power of the other prime factors like 3, 5 and 7 are much lesser than 6.
Statement II alone is sufficient to say that the value of a is 2. The correct answer option is B.
Hope that helps!
