What is the remainder when the product of 171 and 173 is divided by 17?

Question

Examoogle · Accepted Answer

The correct answer is Option B: 3. Find the remainder of each number individually when divided by 17, then multiply those remainders and take the final remainder: $171 = 17 	imes 10 + 1$ (rem 1) and $173 = 17 	imes 10 + 3$ (rem 3).

Step-by-Step Solution & Concepts:
Shortcut Trick: Find the remainder of each number individually when divided by 17, then multiply those remainders and take the final remainder: $171 = 17 	imes 10 + 1$ (rem 1) and $173 = 17 	imes 10 + 3$ (rem 3).

Formula Used: $$(a 	imes b) \pmod{n} = [(a \pmod{n}) 	imes (b \pmod{n})] \pmod{n}$$

Given: Numbers to be multiplied: 171 and 173. Divisor: 17.

Steps: Divide 171 by 17: Calculate the remainder of the first number: $171 = 170 + 1$, so $171 \equiv 1 \pmod{17}$.. Divide 173 by 17: Calculate the remainder of the second number: $173 = 170 + 3$, so $173 \equiv 3 \pmod{17}$.. Apply Modular Multiplication: Multiply the remainders together: $1 	imes 3 = 3$. Since 3 is less than 17, the final remainder is 3.

Answer: B is correct because the product of the individual remainders of 171 (1) and 173 (3) when divided by 17 is 3.

Exam Tip: This modular arithmetic technique is extremely useful for finding the last digits of large powers or solving complex divisibility problems in competitive exams.

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