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Bunuel
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What is the greatest integer, k, such that 5^k is a factor of the prod 24 Jan 2019, 03:29
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

15% (low)

Question Stats:

76% (00:53) correct 24% (01:12) wrong based on 271 sessions

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What is the greatest integer, k, such that 5^k is a factor of the product of the integers from 1 through 24, inclusive?

A 1
B 2
C 3
D 4
E 5
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D
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What is the greatest integer, k, such that 5^k is a factor of the prod 24 Jan 2019, 03:45
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Bunuel
What is the greatest integer, k, such that 5^k is a factor of the product of the integers from 1 through 24, inclusive?

A 1
B 2
C 3
D 4
E 5
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product of the integers from 1 to 24 = 24!

5^k is a factor of 24!.

greatest value of k.

24/5 = 4.

D is the correct answer.
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What is the greatest integer, k, such that 5^k is a factor of the prod 24 Jan 2019, 04:44
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Bunuel
What is the greatest integer, k, such that 5^k is a factor of the product of the integers from 1 through 24, inclusive?

A 1
B 2
C 3
D 4
E 5
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to determine the 5^k factor ; value of k
24!/5 = 4

IMO D
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What is the greatest integer, k, such that 5^k is a factor of the prod 24 Jan 2019, 04:58
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Solution


Given:
    • \(5^k\) is a factor of the product of the integers from 1 through 24, inclusive

To find:
    • The greatest integer value of k

Approach and Working:
    • k = the number of 5’s in 24!• Implies, k = \([\frac{24}{5}] + [\frac{24}{25}] = 4 + 0 = 4\)

Therefore, the maximum possible value of k = 4

Hence, the correct answer is Option D

Answer: D
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What is the greatest integer, k, such that 5^k is a factor of the prod 03 Apr 2025, 08:25
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Bunuel
What is the greatest integer, k, such that 5^k is a factor of the product of the integers from 1 through 24, inclusive?

A 1
B 2
C 3
D 4
E 5
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We want to know how many times the prime number 5 appears in the prime factorization of the product 1 × 2 × 3 × ... × 24 (which is 24!). Every time a number in that range is divisible by 5, it contributes at least one factor of 5. Some numbers might contribute more than one factor if they are divisible by a higher power of 5.
Step 1: Count the numbers from 1 to 24 that are multiples of 5. These numbers are 5, 10, 15, and 20. That gives us 4 factors of 5.
Step 2: Check if any number in the range is a multiple of 25 (since 25 = 52 would contribute an extra factor of 5). In the range from 1 to 24, there are no numbers that are 25 or greater, so none of the numbers provide an extra factor.
So, in total, 5 appears 4 times in the factorization of 24!. This means the greatest integer k such that 5^k divides the product is 4.
The correct answer is 4 (option D).
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