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number properties (700 level question)

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number properties (700 level question)

by Night reader » Sat Oct 30, 2010 11:00 am
The product of the first 7 positive integers is x; x is not a multiple of

(A) 81
(B) 101
(C) 47
(D) 77
(E) 81
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by Rahul@gurome » Sat Oct 30, 2010 11:03 am
Night reader wrote:The product of the first 7 positive integers is x; x is not a multiple of

(A) 81
(B) 101
(C) 47
(D) 77
(E) 81
I guess there is something wrong with the question and/or the options.
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by Night reader » Sat Oct 30, 2010 11:09 am
---
Last edited by Night reader on Sat Oct 30, 2010 11:15 am, edited 1 time in total.
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by Night reader » Sat Oct 30, 2010 11:14 am
Night reader wrote:The product of the first 7 positive integers is x; x is not a multiple of

(A) 81
(B) 101
(C) 47
(D) 77
(E) 81

Recall previous - text of the problem follows

The product of the first 7 positive integers is x; x is not a multiple of

(A) 72
(B) 105
(C) 48
(D) 44
(E) 84
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by Rahul@gurome » Sat Oct 30, 2010 11:22 am
Night reader wrote: keep working on - pick number method should help with the delay of 3-4 minutes (no calculator if one solves hand-mind)
key=basic concepts of the number properties
Well, here is the solution:

x = product of first 7 positive integers = 1*2*3*4*5*6*7 = (2^4)*(3^2)*5*7

(1) x is not multiple of 47 and 101 as they are primes greater than 7.
(2) x is not multiple of 81 as 81 = 3^4 and highest power of 3 in x is 2.
(3) x is not multiple of 77 as 77 = 7*11 and doesn't contain 11.

None of the options are correct and 81 appears twice in the option.
So there must be something wrong!
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by Rahul@gurome » Sat Oct 30, 2010 11:31 am
Night reader wrote:Recall previous - text of the problem follows

The product of the first 7 positive integers is x; x is not a multiple of

(A) 72
(B) 105
(C) 48
(D) 44
(E) 84
Now it looks good. :)

x = product of first 7 positive integers = 1*2*3*4*5*6*7 = (2^4)*(3^2)*5*7

Find the prime factorizations of the options. The one we are searching for should have,
  • (1) A prime factor greater than 7
    or
    (2) Prime factors with exponents greater than that in x. For 2, greater than 4; For 3, greater than 2 etc.
We can avoid these individual factorizations just by noting that 44 = 4*11, where 11 is a prime factor greater than 11. Thus x is not a multiple of 44.

The correct answer is D.
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by pesfunk » Sat Oct 30, 2010 8:53 pm
Very simple...the answer is D

since 77 = 7*11 and 11 is a prime as well as NOT among the first 7 positive integers,
77 wouldn't be able to divide the product completely.
Night reader wrote:The product of the first 7 positive integers is x; x is not a multiple of

(A) 81
(B) 101
(C) 47
(D) 77
(E) 81
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by cauperwood » Mon Nov 01, 2010 12:21 pm
Ohh yeah? Try dividing that product by 81...or 101...or 47.
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