1. Consider the integer A 4444444. Let B denote the sum of the digits of A, let

Question

1. Consider the integer A 4444444. Let B denote the sum of the digits of A, let C denote the sum of the digits of B, and let D denote the sum of the digits of C. Find Here is a strategy for finding D (a) Show that, for any integer N, N the sum of the digits of N (mod 9). Use this property to find the class of D mod 9. (b) Denote the number of digits of A by k. Estimate k and use this estimate to show that B 9k (c) Repeat the previous step to estimate C and D (d) In the previous step you should find that D is very small and knowing the class of A mod 9 should be sufficient to determine D

Explanation / Answer

44447(mod9) so that 444431(mod9) and then get 444444447(mod9).

Then use the fact that for any integer N, the sum of the digits of N is equivalent to N(mod9).

Finally use logs to base 10 to get a limit on the size of A, hence B .

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