Question

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

Answer 1

Correct Answer: 3000

 To find the total number of 4-digit numbers whose greatest common divisor (GCD) with 54 is 2, follow these steps:

1. First, identify the range of 4-digit numbers, which is from 1000 to 9999.
2. Next, break down 54 into its prime factors: 54 = 2 × 3³.
3. The GCD of a number with 54 being 2 means the number must not have 3 as a factor but must have 2 as a factor.
4. To ensure a number is divisible by 2, the number must be even. For an even number not to be divisible by 3, the number must not satisfy divisibility conditions for 3.
5. Total even numbers from 1000 to 9999: The sequence is 1000, 1002, ..., 9998. This is an arithmetic sequence with the first term a₁ = 1000, common difference d = 2, and nth term a_n = 9998. To find n:

a_n = a₁ + (n-1)d => 9998 = 1000 + (n-1)×2 => n = 4500.

1. Total numbers divisible by both 2 and 3 (hence divisible by 6): Sequence is 1002, 1008, ..., 9996, where the first term is 1002, d = 6, a_n = 9996. Solve for n:

a_n = a₁ + (n-1)d => 9996 = 1002 + (n-1)×6 => n = 1500.

1. Subtract numbers divisible by 6 from the total even numbers: 4500 - 1500 = 3000.
2. Thus, the count of 4-digit numbers where the GCD with 54 is 2 is 3000.

The result, 3000, fits the expected range.