Which of the following numbers, when divided by 42, yields no fraction?

A number is divisible by 2 if its last digit is even.

• (A) 26565: Ends in 5 (odd). Not divisible by 2.
• (B) 26574: Ends in 4 (even). Potentially divisible by 2.
• (C) 26586: Ends in 6 (even). Potentially divisible by 2.
• (D) 26600: Ends in 0 (even). Potentially divisible by 2. Option (A) is eliminated.

A number is divisible by 3 if the sum of its digits is divisible by 3.

• (B) 26574: Sum of digits = $2+6+5+7+4 = 24$. Since 24 is divisible by 3, 26574 is divisible by 3. (Since it's also divisible by 2, it's divisible by 6).
• (C) 26586: Sum of digits = $2+6+5+8+6 = 27$. Since 27 is divisible by 3, 26586 is divisible by 3. (Since it's also divisible by 2, it's divisible by 6).
• (D) 26600: Sum of digits = $2+6+6+0+0 = 14$. Since 14 is not divisible by 3, 26600 is not divisible by 3. Option (D) is eliminated.

We will perform long division for these two numbers.

For option (B) 26574:

$$\frac{26574}{42}$$

Divide 265 by 42: $265 = 6 \times 42 + 13$.
Bring down 7, making 137.
Divide 137 by 42: $137 = 3 \times 42 + 11$.
Bring down 4, making 114.
Divide 114 by 42: $114 = 2 \times 42 + 30$.
Since there is a remainder of 30, 26574 is not perfectly divisible by 42.

For option (C) 26586:

$$\frac{26586}{42}$$

Divide 265 by 42: $265 = 6 \times 42 + 13$.
Bring down 8, making 138.
Divide 138 by 42: $138 = 3 \times 42 + 12$.
Bring down 6, making 126.
Divide 126 by 42: $126 = 3 \times 42 + 0$.
Since the remainder is 0, 26586 is perfectly divisible by 42.

Thus, 26586 is the number that, when divided by 42, yields no fraction.

The final answer is $\boxed{\text{C) 26586}}$.