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The number 5^7
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25 Aug 2025, 06:55
25 Aug 2025, 06:55
Expert Reply
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Question Stats:
100% (00:27) correct
0% (00:00) wrong based on 4 sessions
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The number $\(5^7\)$ must be divisible by all of the following EXCEPT:
(A) 625
(B) 125
(C) 25
(D) 15
(E) 5
(A) 625
(B) 125
(C) 25
(D) 15
(E) 5
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The number 5^7
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29 Aug 2025, 08:51
29 Aug 2025, 08:51
Expert Reply
The correct answer is (D) 15.
To figure this out, you can break down the number $\(5^7\)$ into its prime factors. The number $\(5^7\)$ is simply 5 multiplied by itself 7 times. Its only prime factor is 5 .
Now, let's look at each option:
- (A) 625: This can be written as $\(5 \times 5 \times 5 \times 5=5^4\)$. Since $\(5^7\)$ contains four factors of 5 , it is divisible by $\(5^4\)$.
- (B) 125: This is $\(5 \times 5 \times 5=5^3\)$. Since $\(5^7\)$ contains three factors of 5 , it is divisible by $\(5^3\)$.
- (C) 25: This is $\(5 \times 5=5^2\)$. Since $\(5^7\)$ contains two factors of 5 , it is divisible by $\(5^2\)$.
- (D) 15: The prime factors of 15 are $\(3 \times 5\)$. For a number to be divisible by 15 , it must be divisible by both 3 and 5 . The number $5^7$ has no factor of 3 , so it is not divisible by 15 .
- (E) 5 : This is a factor of $\(5^7\)$.
Therefore, the only number that $\(5^7\)$ is not divisible by is 15 .
To figure this out, you can break down the number $\(5^7\)$ into its prime factors. The number $\(5^7\)$ is simply 5 multiplied by itself 7 times. Its only prime factor is 5 .
Now, let's look at each option:
- (A) 625: This can be written as $\(5 \times 5 \times 5 \times 5=5^4\)$. Since $\(5^7\)$ contains four factors of 5 , it is divisible by $\(5^4\)$.
- (B) 125: This is $\(5 \times 5 \times 5=5^3\)$. Since $\(5^7\)$ contains three factors of 5 , it is divisible by $\(5^3\)$.
- (C) 25: This is $\(5 \times 5=5^2\)$. Since $\(5^7\)$ contains two factors of 5 , it is divisible by $\(5^2\)$.
- (D) 15: The prime factors of 15 are $\(3 \times 5\)$. For a number to be divisible by 15 , it must be divisible by both 3 and 5 . The number $5^7$ has no factor of 3 , so it is not divisible by 15 .
- (E) 5 : This is a factor of $\(5^7\)$.
Therefore, the only number that $\(5^7\)$ is not divisible by is 15 .
gmatclubot
The number 5^7 [#permalink]
29 Aug 2025, 08:51
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