What is the remainder when 91 x 92 x 93 x 94 x 95 x 96 x 97 x 98 x 99 is divided by 1261?

Key fact: The product is (91\times 92\times \cdots \times 99), i.e. 9 consecutive integers.
Factorize divisor: (1261 = 13\times 97).
Check divisibility by 13: Among 91 to 99, (91=13\times 7), so the product is divisible by 13.
Check divisibility by 97: Among 91 to 99, (97) itself is present, so the product is divisible by 97.
Since the product is divisible by both 13 and 97, it is divisible by (1261). Hence the remainder is 0.

Option analysis:

• (a) 3 — Incorrect; product is exactly divisible by 1261.
• (b) 2 — Incorrect; product is exactly divisible by 1261.
• (c) 1 — Incorrect; product is exactly divisible by 1261.
• (d) 0 — Correct; divisor factors (13\times 97) both occur in the product.