Difficulty: Easy
Correct Answer: 3
Explanation:
Introduction / Context:
This question tests your understanding of remainders and modular arithmetic. It uses a larger divisor first and then asks for the remainder with a factor of that divisor, which is a common pattern in remainder based aptitude problems.
Given Data / Assumptions:
Concept / Approach:
We can express N in terms of quotient and remainder using the division algorithm: N = 138k + 26 for some integer k. Then we reduce this expression modulo 23. Since 138 is a multiple of 23, the term 138k will vanish modulo 23, leaving only the remainder from the constant part.
Step-by-Step Solution:
Step 1: Write N in division algorithm form: N = 138k + 26, where k is an integer.Step 2: Notice that 138 = 23 * 6, so 138 is divisible by 23.Step 3: Consider N modulo 23: N mod 23 = (138k + 26) mod 23.Step 4: Since 138k is a multiple of 23, 138k mod 23 = 0.Step 5: Therefore, N mod 23 = 26 mod 23.Step 6: Compute 26 divided by 23: 26 = 23 * 1 + 3, so the remainder is 3.Step 7: Hence, the remainder when N is divided by 23 is 3.
Verification / Alternative check:
Take an example value: choose k = 1. Then N = 138 * 1 + 26 = 164. Now 164 ÷ 23 = 7 remainder 3, which matches our answer.Any other integer value of k will shift N by multiples of 138, which are also multiples of 23, so the remainder modulo 23 always stays 3.
Why Other Options Are Wrong:
Remainders 4, 6 and 1 would come from incorrect arithmetic when reducing 26 modulo 23.They usually reflect mistakes like subtracting the wrong multiple of 23 or confusing quotient with remainder.
Common Pitfalls:
Forgetting that 138 is a multiple of 23 may lead to unnecessary long division.Some learners mistakenly try to reconstruct N explicitly instead of working symbolically with k.Recognising multiples of the smaller divisor inside the larger divisor simplifies the problem significantly.
Final Answer:
The remainder when the number is divided by 23 is 3.
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