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How much zeros has the number $1000!$ at the end? If a number ends with n n zeros than it is divisible by 10n 10 n, that is 2n5n 2 n 5 n A factorial clearly has more 2 2 s than 5 5 s in its factorization so you only need to count how many 5 5 s are there in the factorization of 1000! 1000!
Find the number of times - Mathematics Stack Exchange Question: Find the number of times 5 5 will be written while listing integers from 1 1 to 1000 1000 Now, it can be solved in this fashion The numbers will be of the form: 5xy, x5y, xy5 5 x y, x 5 y, x y 5 where x, y x, y denote the two other digits such that 0 ≤ x, y ≤ 9 0 ≤ x, y ≤ 9 So, x, y x, y can take 10 10 choice each
Is there a faster way to count jigsaw puzzle pieces? The old 1000 1000 -piece jigsaw puzzles in my home cannot be tested for completeness by mere counting Alas, my children have on occasion mixed up pieces from different puzzles, and having 1000 1000 pieces in a box does not guarantee that the puzzle is complete
Find $A^{1000}$ by using Cayley-Hamilton Theorem Find A1000 A 1000 by using the Cayley-Hamilton theorem I find the characteristic polynomial by P(A) = −A3 + 2A2 = 0 P (A) = A 3 + 2 A 2 = 0 (by Cayley-Hamilton) but I don't see how to find A1000 A 1000 by this characteristic polynomial