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04 Jul 2024, 01:07
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B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
55%
(hard)
Question Stats:
61% (01:46) correct
39%
(02:06)
wrong
based on 94
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What is the highest power of 3 available in the expression 58! - 38!?
A. 16
B. 17
C. 18
D. 19
E. 20
A. 16
B. 17
C. 18
D. 19
E. 20
04 Jul 2024, 01:12
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Bunuel
What is the highest power of 3 available in the expression 58! - 38!?
A. 16
B. 17
C. 18
D. 19
E. 20
58! - 38! =A. 16
B. 17
C. 18
D. 19
E. 20
= 38!(58 * 57 * 56 * ... * 39 - 1)
58 * 57 * 56 * ... * 39 is a multiple of 3, thus 58 * 57 * 56 * ... * 39 - 1 is 1 less than a multiple of 3, so not a multiple of 3 and thus will not contribute any 3's to the entire product.
The highest integer power of 3 in 38! is 38/3 + 38/9 + 38/27 = 12 + 4 + 1 = 17 (remember we take only the quotients of the divisions, that is 38/3 would give 12, not 12 2/3).
Answer: A.
- For theory on the topic discussed in the question check: Number Properties
For similar questions check the following topics from our Special Questions Directory:
12. Trailing Zeros
13. Power of a number in a factorial
Hope it helps.
04 Jul 2024, 01:21
Kudos
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Bunuel
Bunuel
What is the highest power of 3 available in the expression 58! - 38!?
A. 16
B. 17
C. 18
D. 19
E. 20
58! - 38! =A. 16
B. 17
C. 18
D. 19
E. 20
= 38!(58 * 57 * 56 * ... * 39 - 1)
58 * 57 * 56 * ... * 39 is a multiple of 3, thus 58 * 57 * 56 * ... * 39 - 1 is 1 less than a multiple of 3, so not a multiple of 3 and thus will not contribute any 3's to the entire product.
The highest integer power of 3 in 38! is 38/3 + 38/9 + 38/27 = 12 + 4 + 1 = 17 (remember we take only the quotients of the divisions, that is 38/3 would give 12, not 12 2/3).
Answer: A.
- For theory on the topic discussed in the question check: Number Properties
For similar questions check the following topics from our Special Questions Directory:
12. Trailing Zeros
13. Power of a number in a factorial
Hope it helps.
Or:
\(58! - 38! =\)
\(= 3^{27}*x -3^{17}*y =\)
\(= 3^{17}(3^{10}*x -y) =\)
\(= 3^{17}(multiple \ of \ 3 - NOT \ a \ multiple \ of \ 3) =\)
\(= 3^{17}(NOT \ a \ multiple \ of \ 3)\)
Answer: B.
