JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Permutations and Combinations

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is ______

Answer: 1000

Solution

Any such number is a multiple of

3

, so write

n = 3 k

. Then

g cd (n, 18) = g cd (3 k, 3 \cdot 6) = 3 g cd (k, 6)

and the requirement

g cd (n, 18) = 3

is exactly

g cd (k, 6) = 1

.
The four-digit multiples of

3

are

1002, 1005, \dots, 9999

, corresponding to

k = 334, 335, \dots, 3333

, a block of

3333 - 334 + 1 = 3000

consecutive integers. Among any six consecutive integers exactly two are coprime to

6

, and

3000

is a multiple of

6

, so the count is

3000 \times \frac{2}{6} = 1000

There are

1000

such four-digit numbers.

JEE Main · 2021 · 26 Feb 2021 · Shift 2 · Integer answer