### Brain Teasers

# Threes, Fives and Sevens

Which is the smallest natural number that satisfies the following conditions:

(1) Its digits consist only of, and only of 3's, 5's and 7's.

(2) Its digital sum is divisible by 3, 5 and 7.

(3) The number itself is divisible by 3, 5 and 7.

(1) Its digits consist only of, and only of 3's, 5's and 7's.

(2) Its digital sum is divisible by 3, 5 and 7.

(3) The number itself is divisible by 3, 5 and 7.

### Hint

It's easier than some of you might think.### Answer

From condition 2, its digital sum is at least 105.From condition 3, it must end with 5, but its divisibility by 3 is guaranteed, so we only have to check 7.

From condition 1 and 2, it has an odd number of digits. 105/7=15, so it has more than 15, and at least 17 digits. We can have:

(1) 13 7's, 1 5, and 3 3's;

(2) 12 7's, 3 5's, and 2 3's;

(3) 11 7's, 5 5's, and 1 3.

In the second case, we verify that neither 33557777777777775 nor 33575777777777775 is divisible by 7, but 33577577777777775 is divisible by 7 (the quotient is 4796796825396825.)

In the first case, the only number they make that is less than 33577577777777775 is 33377777777777775, which is not divisible by 7.

In the third case, they can't make a number less than 33577577777777775.

Therefore, the number we're looking for is 33577577777777775.

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