Answer to Question #302189 in Combinatorics | Number Theory for ABRAHAM SALOMO P

Question

Qn 1. An integer N has digital representation a1a2a3. Moreover,

ˆ None of the digits a1, a2, or a3 and none of the numbers with digital

representation a1a2, a1a3, or a2a3 is divisible by 3.

ˆ N is odd.

ˆ N is divisible by 9.

ˆ a1 ≥ a2 ≥ a3.

Determine all possible numbers N.

Answer 1

 $N$ is odd , and none of the digits $a_1, a_2,$ or $a_3$ is divisible by $3$

$$a_3=1, 5, 7$$

$N$ is divisible by 9

$$a_1+a_2+a_3=9k, k=1,2$$

Let $a_3=1.$

$a_1\ge a_2 \ge a_3,$ none of the digits $a_1, a_2,$ or $a_3$ is divisible by $3$ , and none of the numbers with digital representation $a_1a_2, a_1a_3,$ or $a_2a_3$ is divisible by 3.

$$a_2=1, a_3=7$$

Or

$$a_2=4, a_3=4$$

Let $a_3=5.$

$a_1\ge a_2 \ge a_3,$ none of the digits $a_1, a_2,$ or $a_3$ is divisible by $3$ , and none of the numbers with digital representation $a_1a_2, a_1a_3,$ or $a_2a_3$ is divisible by 3.

$$a_2=5, a_3=8$$

Let $a_3=7.$

$a_1\ge a_2 \ge a_3,$ none of the digits $a_1, a_2,$ or $a_3$ is divisible by $3$ , and none of the numbers with digital representation $a_1a_2, a_1a_3,$ or $a_2a_3$ is divisible by 3.

It is impossible to find the number.

The possible numbers are $441, 711, 855.$