Brilliant question

Let A be a number with 2001 digits such that A is a multiple of 10! Let B be the digit sum of A, C be the digit sum of B, and D be the digit sum of C. What is the unit’s digit of D?

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Let $S(n)$ denote the sum of the digits of $n$ . Since $A$ has $2001$ digits, $B = S(A) \le 9 \cdot 2001 = 18009$ .

Since $B$ has at most $5$ digits, $C = S(B) \le 9 \cdot 5 = 45$ . By inspection, $D = S(C) \le 3 + 9 = 12$ .

Since $S(n) \equiv n \pmod{9}$ for any integer $n$ , we have $D \equiv C \equiv B \equiv A \equiv 10! \cdot m \equiv 0 \pmod{9}$ .

Trivially, $D \ge 1$ . Since $1 \le D \le 12$ and $D \equiv 0 \pmod{9}$ , we have $D = 9$ , and thus, the unit digit is $9$ .

Jimmy Kariznov - 7 years, 10 months ago

Awesome buddy.THNX

alpha beta - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$