JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Permutations and Combinations

The total number of four digit numbers such that each of the first three digits is divisible by the last digit, is equal to ______.

Answer: 1086

Solution

Let the four-digit number be

ab c d

where

a, b, c

are each divisible by

d

.

For each value of

d

, count the choices for

a, b, c

$d = 1$ : All digits 1-9 for $a$ , 0-9 for $b, c$ → $9 \times 10 \times 10 = 900$
$d = 2$ : $a \in {2, 4, 6, 8}$ , $b, c \in {0, 2, 4, 6, 8}$ → $4 \times 5 \times 5 = 100$
$d = 3$ : $a \in {3, 6, 9}$ , $b, c \in {0, 3, 6, 9}$ → $3 \times 4 \times 4 = 48$
$d = 4$ : $a \in {4, 8}$ , $b, c \in {0, 4, 8}$ → $2 \times 3 \times 3 = 18$
$d = 5$ : $a = 5$ , $b, c \in {0, 5}$ → $1 \times 2 \times 2 = 4$
$d \in {6, 7, 8, 9}$ : $a = d$ , $b, c \in {0, d}$ → $4 \times (1 \times 2 \times 2) = 16$

Total:

900 + 100 + 48 + 18 + 4 + 16 = 1086

JEE Main · 2022 · 29 Jun 2022 · Shift 2 · Integer answer