Oddly Divisible

Algebra Level 3

Let $x$ be a three-digit integer, $s$ be the sum of its digits, and $p$ be the product of its digits. And that

$x$ is divisible by $s$ and $p$ .
$p$ is divisible by $s$ .
$p$ and $s$ are odd numbers.

What is $x$ ?

The answer is 735.

3 solutions

Fletcher Mattox
Oct 12, 2020

Feeling redundant this morning. :)

from numpy import prod

for n in range(100, 1000):
    s = sum(list(map(int, list(str(n)))))
    p = prod(list(map(int, list(str(n)))))
    if n%2 == 1 and p != 0 and n%p == n%s == p%s == 0:
        print(n)

Chew-Seong Cheong
Oct 12, 2020

Since brutal force seemed inevitable, I used Python coding to solve it.

Anh Khoa Nguyễn Ngọc
Oct 12, 2020

The third condition indicated that all three digits of $x$ are odd, so it narrowed down the search to $125$ numbers.

The first and second showed that $x$ must be a composite number thus the searching list is now only $82$ numbers.

Using logic and a trick called brute force, we have the answer is $\blue{\boxed{735}}$ .

Oddly Divisible

The answer is 735.

3 solutions

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