Multiply and Add

Logic Level 4

Given any positive integer $n$ , let $p(n)$ be the product of the non-zero digits of $n$ . If $n$ has only one digit, then $p(n)$ is equal to that digit.

$Let: S = p(1) + p(2) + p(3) +\ldots + p(999)$ .

What is the largest prime factor of $S$ ?

The answer is 103.

1 solution

Chew-Seong Cheong
Jun 4, 2015

$\begin{aligned} S & = p(1)+p(2)+p(3)+...+p(999) \\ & = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 \\ & \quad \quad +1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 \\ & \quad \quad \quad + 2 (1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) \\ & \quad \quad \quad \quad + 3 (1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) +... \\ & \quad \quad \quad \quad \quad ... + 9\times 9(1 + 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) \\ & = -1 + (1)(46) + (1)(46) + 2(46) + 3(46) +...\\ & \quad \quad \quad ... + 9(7)(46) + 9(8)(46) + 9(9)(46) \\ & = -1 + (46)(46)(46) = 46^3 -1 = 97335 = 3^3\dot{}5\dot{}7\dot{}103 \end{aligned}$

Therefore, the largest prime factor of $S$ is $\boxed{103}$ .

Moderator note:

Find a one-paragraph solution to show that the sum is equal to $46^3 - 1$ .

What u meant by one paragraph solution? Is this provided solution long?

Ravi Dwivedi - 5 years, 5 months ago

Multiply and Add

The answer is 103.

1 solution

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