JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Permutations and Combinations

The number of 3-digit odd numbers, whose sum of digits is a multiple of 7, is ______.

Answer: 63

Solution

A 3-digit odd number has form

x y z

where

z \in {1, 3, 5, 7, 9}

x \in {1, 2, ..., 9}

y \in {0, 1, ..., 9}

.

We need

x + y + z \equiv 0 (mod 7)

, so

x + y + z \in {7, 14, 21}

(since max sum is

9 + 9 + 9 = 27

).

For each value of

z

, we need

x + y \in {6, 4, 2, 13, 11, 9, 7, 5, 20, 18, 16, 14, 12}

(depending on

z

).

For

z = 1

x + y \in {6, 13, 20}

Total: 12

For

z = 3

x + y \in {4, 11, 18}

Total: 13

For

z = 5

x + y \in {2, 9, 16}

Total: 14

For

z = 7

x + y \in {0, 7, 14}

Total: 12

For

z = 9

x + y \in {5, 12}

(21-9=12, 14-9=5)

Total: 12

Grand total:

12 + 13 + 14 + 12 + 12 = 63

JEE Main · 2022 · 25 Jun 2022 · Shift 1 · Integer answer