What is the remainder when 85 × 87 × 89 × 91 × 95 × 96 is divided by 100 ?

(a) 0 ✓ UPSC's answer
(b) 1
(c) 2
(d) 4

Why the answer is (a)

• To find the remainder when a number is divided by 100, we need to determine the last two digits of the product.

• The product is 85 × 87 × 89 × 91 × 95 × 96.

• Observe that 85 ends in 5 and 96 ends in 6. The product of any integer ending in 5 and any even integer ends in 0.

• Specifically, 85 × 96 = 8160, which ends in 60.

• Since one factor in the product (85 × 96) is a multiple of 10, the entire product is a multiple of 10.

• Furthermore, 85 is a multiple of 5 and 96 is a multiple of 2, so their product is a multiple of 10. However, to be a multiple of 100, we need two factors of 5 and two factors of 2. 85 = 5 × 17, 95 = 5 × 19, 96 = 32 × 3. The product contains 5 × 5 × 32 × ... = 25 × 32 × ... = 800 × ... which is divisible by 100.

• Therefore, the last two digits are 00, and the remainder is 0.

Why the other options are wrong

(b) 1: The product contains factors 85 and 95 (both multiples of 5) and 96 (a multiple of 4), making the total product divisible by 100, so the remainder cannot be 1.
(c) 2: The product is divisible by 100 because it contains at least two factors of 5 (from 85 and 95) and at least two factors of 2 (from 96), resulting in a remainder of 0, not 2.
(d) 4: The product is divisible by 100 as established by the presence of sufficient factors of 2 and 5, so the remainder is 0, not 4.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.