Prime Power in Factorial

Number Theory Level 3

Find the largest positive integer k k such that 1 1 k 11^k divides 1000 ! 1000! .

Notation: ! ! is the factorial notation. For example, 8 ! = 1 × 2 × 3 × ⋯ × 8 8! = 1\times2\times3\times\cdots\times8 .


The answer is 98.

Muhammad Rasel Parvej by Muhammad Rasel Parvej , 27, Bangladesh

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1 solution

Phillip Temple
Nov 29, 2016

k will be the number of 11s being multiplied inside the factorial. All values 11 ⋅ 1 11\cdot1 , 11 ⋅ 2 11\cdot2 ... 11 ⋅ n 11\cdot n until 11 ⋅ n 11\cdot n > 1000 are to be considered. 11 ⋅ 90 11\cdot90 (990) is the largest scalar of 11 within this range, so 90 11s are in the factorial. Next, cases where there are multiple 11s (e.g. 11 ⋅ 11 11\cdot11 ) are taken into account. Values from 11 ⋅ 11 11\cdot11 , 11 ⋅ 22 11\cdot22 to 11 ⋅ 88 11\cdot88 produce 8 more 11s. Since the next case ( 11 ⋅ 121 11\cdot121 ) is larger than 1000, the answer is 90 +8 = 98

5 Helpful 0 Interesting 0 Brilliant 0 Confused

Since the next case ( 11 ⋅ 121 11 \cdot 121 ) is larger than 1000.

I think you meant 11 ⋅ 99 11\cdot 99 ? 88 → 99 88\rightarrow 99 ?

Also, I think it'll be clearer if you say values from 1 1 2 ⋅ 1 , 1 1 2 ⋅ 2 , … 11^2\cdot 1, 11^2 \cdot 2, \dots instead of 11 ⋅ 11 , 11 ⋅ 22 , … 11\cdot 11, 11\cdot 22, \dots as the former gives a stronger emphasis on the second 11. What do you think?

Christopher Boo - 4 years, 6 months ago

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