The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is ______.

Question

The total number of 4-digit numbers whose greatest common divisor with 54 is 2, is \_\_\_\_\_\_.

Answer: 3000

Accepted Answer

We need $N$ divisible by $2$ but not by $3$ (since $54=2	imes27$, $\gcd(N,54)=2$ requires $2\mid N$ and $3
mid N$).

4-digit numbers: $9000$ total.

Divisible by 2: $9000/2=4500$. Divisible by 6 (2 and 3): $9000/6=1500$.

$N = 4500-1500 = 3000$

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