The numerals 10000, 121, 100, 31, 24, P and 20 are written in bases 2, 3, 4,…
2022
The numerals 10000, 121, 100, 31, 24, P and 20 are written in bases 2, 3, 4, 5, 6, 7 and 8 respectively. If each numeral represents the same decimal number x, what is P?
Answer: A. 22 — ConceptIn a positional numeral system of base b, a two-digit numeral uv represents u × b + v. More generally, each digit is multiplied by the corresponding…
- A.
22
- B.
21
- C.
23
- D.
16
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Correct answer: A
Concept
In a positional numeral system of base b, a two-digit numeral uv represents u × b + v. More generally, each digit is multiplied by the corresponding power of the base.
To compare representations in different bases, first convert a known representation to decimal and then express that decimal value in the required base.
Application
The listed bases are 2 through 8. Using the final representation, 208 = 2 × 8 + 0 = 16, so x = 16.
The missing numeral P is the base-7 representation of 16.
Divide 16 by 7: 16 = 2 × 7 + 2. The quotient gives the sevens digit and the remainder gives the units digit, so P = 227.
Cross-check
Converting back, 227 = 2 × 7 + 2 = 16, matching 100002, 1213, 1004, 315, 246 and 208.
Result
Therefore, P = 22.