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If a and n are positve integers > 1, and the product of the first 8 integers is a multiple of a^n, what is a?
If you know that:
a) a^n = 64
b) n = 6
A) I can tell the answer from eqn (a) alone
B) I can tell the answer from eqn (b) alone
C) I need both eqn (a) and eqn(b) to tell the answer
D) I can't tell from either one.
Please let me know how you came to your answer.
Other than just trial and error, is there a way to figure it out? Without multiplying out 8!.
1 Answer
- Don E KnowsLv 61 decade agoFavorite Answer
Answer C.
You can get ONE answer that works from equation a alone : a = 2 and n = 6 will give you 2^6 = 64. But other numbers based on logs will also solve this.
But when you set n to 6, then there are still two values for a that will work : a = 2 or a = -2.
But we specified that a must be positive. So only 2 is possible -- therefore answer C.
As for determining that 8! has 64 as a factor :
8! has 2, 4 and 8 in it. And 2 x 4 x 8 = 64.
To find out what multiple of 64 that 8! is :
Take out these factors, and multiply the rest.
1 x 3 x 5 x 6 x 7 = 630
So 630 x 64 = 8! = 40320
