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22 Jan 2021, 05:45
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E
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:
57% (01:38) correct
43%
(01:48)
wrong
based on 212
sessions
History
Date
Time
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Not Attempted Yet
What is the greatest integer value of n, such that \(\frac{99!-98!}{7^n}\) is an integer ?
A. 8
B. 15
C. 16
D. 17
E. 18
A. 8
B. 15
C. 16
D. 17
E. 18
22 Jan 2021, 16:50
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IMO it is B. 98!(99-1)/7 =98!(98)/7 there is 10 multiples of seven through 70. 77 84 91 98 98 (there is 2 98) total of 15.
22 Jan 2021, 18:10
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Bunuel
\(99!-98! = 99*(98!) - 98! = (99 - 1) * 98! = 98 * 98!\). We want to count how many multiples of seven there are in \(98 * 98!\).
Let's focus on"98!" first, we know 98 = 7*14 so we can begin with 14 multiples of seven, this is what I call the first layer. Next, we can count the second layer by searching for multiples of 49, which are 49 and 98. Both of these each have an extra multiple of 7 since they are multiples of 49. Therefore 98! includes 14+2 = 16 multiples of seven. And finally, 98 itself has 2 multiples of seven so in total we have 18.
Ans: E
