WAEC 2017 Mathematics · Question 24

In what number base was the addition 1 + nn = 100, where n > 0, done?

Options

A.n - 1
B.n + 1
C.n
D.n + 2

Answer: B. n + 1

Explanation written with AI and checked against the answer key

Analyze the question: Here "nn" is a two-digit numeral in some base b, where each digit is n. We need the base b in which 1 + nn = 100. Concept: In base b, the numeral nn means n·b + n = n(b + 1), and 100 means b². Solution: 1 + n(b + 1) = b² Rearranging: b² − n(b + 1) − 1 = 0 Try b = n + 1, so n = b − 1: n(b + 1) = (b − 1)(b + 1) = b² − 1 Then 1 + (b² − 1) = b², which equals 100 in base b. ✓ Check with an example: In base 10 the addition is 1 + 99 = 100. Here n = 9, so the base is n + 1 = 10. ✓ Another example: in base 8, n = 7, and 1 + 77 = 100 (base 8). Indeed 1 + 63 = 64. ✓ The largest digit in any base b is b − 1, so nn is the largest two-digit number. Adding 1 to it always gives 100. This means the base is n + 1. Answer: B (n + 1) Tip: Adding 1 to the largest two-digit number in any base always gives 100 in that base, just as 99 + 1 = 100 in base 10. The base is one more than the biggest digit.

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