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

Step-by-step solution

Four-digit numbers are 1000 , 1001 , . . . , 9999 Total number of four digit numbers are 9000 . Now, 54 = 2 × 3 3 So, for getting gcd with 2 , we must have numbers multiple of 2 but not of 3 . Multiple of 2 are 1000 , 1002 , . . . . . , 9998 Let total numbers be m , then 9998 = 1000 + m - 1 2 ⇒ m = 4500 Total numbers are 4500 . Multiple of 6 are 1002 , 1008 . . . . , 9996 Let total numbers be n , then 9996 = 1002 + n - 1 6 ⇒ n = 1500 So, required number is = 4500 - 1500 = 3000