What is the remainder when 51 × 27 × 35 × 62 × 75 is divided by 100?

To find the remainder when 51 x 27 x 35 x 62 x 75 is divided by 100, we need to find (51 x 27 x 35 x 62 x 75) mod 100.

1. Identify factors of 100: 100 = 2 x 2 x 5 x 5.
2. Look for factors of 2 and 5 in the given numbers:
   • 35 = 5 x 7
   • 62 = 2 x 31
   • 75 = 3 x 25 = 3 x 5 x 5
3. Extract common factors that make up 100 or multiples of 100: From the numbers 35, 62, and 75, we can extract:
   • One '2' from 62.
   • One '5' from 35.
   • Two '5's from 75. Multiplying these extracted factors: 2 x 5 x 5 x 5 = 250. This means the entire product is a multiple of 250.
4. Rewrite the product: The product P = 51 x 27 x 35 x 62 x 75 can be written as P = K x 250, where K is the product of the remaining factors: K = 51 x 27 x 7 x 31 x 3.
5. Find the remainder of P when divided by 100: We need to find (K x 250) mod 100. First, find 250 mod 100: 250 = 2 x 100 + 50, so 250 mod 100 = 50. Therefore, (K x 250) mod 100 = (K x 50) mod 100.
6. Determine if K is even or odd: K = 51 x 27 x 7 x 31 x 3. All these numbers are odd. The product of odd numbers is always odd. So, K is an odd number.
7. Calculate the final remainder: If K is odd, let K = 2m + 1 for some integer m. Then (K x 50) mod 100 = ((2m + 1) x 50) mod 100 = (100m + 50) mod 100. Since 100m is a multiple of 100, (100m + 50) mod 100 = 50.

Alternatively, we can multiply and take the remainder step-by-step: (51 x 27) mod 100 = 1377 mod 100 = 77 (35 x 62) mod 100 = 2170 mod 100 = 70 Now we have (77 x 70 x 75) mod 100. (77 x 70) mod 100 = 5390 mod 100 = 90 Now we have (90 x 75) mod 100. (90 x 75) = 6750. 6750 mod 100 = 50.

Both methods yield the same result.

The final answer is A) 50.