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(1) What is the last nonzero digit of 35!?
(2) Find the last nonzero digit of 35!.
[A math quiz for Japanese high school students. (translation mine)]
I assume #1 is natural. Is #2 natural?


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Yes, #2 is also natural. I'm not sure I understand the question, though. On the face of it, the last nonzero (or nonzero) digit of 35 is the 5 at the end of it. There aren't any zeros (which is what I assume is meant by "zero digit") in 35. There's a three and a five and that's all.
Could it be that the question really is "What is the last digit of 35?" without the word or concept "nonzero"?


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tautophile wrote:Yes, #2 is also natural. I'm not sure I understand the question, though. On the face of it, the last nonzero (or nonzero) digit of 35 is the 5 at the end of it. There aren't any zeros (which is what I assume is meant by "zero digit") in 35. There's a three and a five and that's all. Thanks, tautophile.35! = 35 factorial = 1×2×3×···×33×34×35 The answer is 2. 35! = 10333147966386144929666651337523 200000000


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Reiko07 wrote:tautophile wrote:Yes, #2 is also natural. I'm not sure I understand the question, though. On the face of it, the last nonzero (or nonzero) digit of 35 is the 5 at the end of it. There aren't any zeros (which is what I assume is meant by "zero digit") in 35. There's a three and a five and that's all. Thanks, tautophile.35! = 35 factorial = 1×2×3×···×33×34×35 The answer is 2. 35! = 10333147966386144929666651337523 200000000


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The first sentence can be OK if you add anything else between the factor symbol and a question mark. I think that, for the avoidance of a doubt, it may be helpful to include anything else for the person who understands what mathematical factor is if the same person also knows some supplemental mathematical operators. I had seen, perhaps once, the sign > or, < included below the question mark.


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taurine wrote:The first sentence can be OK if you add anything else between the factor symbol and a question mark. I think that, for the avoidance of a doubt, it may be helpful to include anything else for the person who understands what mathematical factor is if the same person also knows some supplemental mathematical operators. I had seen, perhaps once, the sign > or, < included below the question mark. Thanks, taurine. How about the following version? (3) What is the last nonzero digit of 35! (= 35 factorial)?


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Oh, I see....I didn't notice the ! that made the question refer to 35! or 35factorial. That makes all the difference in the world. The question is about the last nonzero digit of 35!, not of 35, and it now makes sense. I assume that the value for 35! (that is, 10,333,147,966,386,144,929,666,651,337,523,200,000,000) has been correctly calculated. I checked that goggling a table of factorials that goes well beyond 35! and yes, that's the right number. (I'm glad I didn't have to calculate it myself.)
I know that N! is meant to be read and spoken as "N factorial", but I've been tempted from time to time to say "N"a very loud, emphatic N such as might be written nonmathematically as "N!"instead.


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tautophile wrote:I've been tempted from time to time to say "N"a very loud, emphatic N such as might be written nonmathematically as "N!"instead. Ah, but that would be pronounced "N exclam".


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tautophile wrote:I checked that goggling a table of factorials that goes well beyond 35! and yes, that's the right number. (I'm glad I didn't have to calculate it myself.)
It's a "No Calculator Allowed" quiz. I used WolframAlpha to check my answer. WolframAlpha – 35!Here's my calculation: 35! = 2³²×3¹⁵×5⁸×7⁵×11³×13²×17²×19×23×29×31 = (2²⁴×3¹⁵×7⁵×11³×13²×17²×19×23×29×31)×100000000 In modulo 10, 2²⁴×3¹⁵×7⁵×11³×13²×17²×19×23×29×31 ≡ 6×7×7×1×9×9×9×3×9×1 ≡ 2×3×1×7 ≡ 2


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