With reference to different number systems, which of the following statements…
2024
With reference to different number systems, which of the following statements is true?
Answer: B. (123)5 = (212)4 — Concept: in a positional number system with base b, the digit positions carry the weights b0, b1, b2, b3 and so on, counting from the right. The value of a…
- A.
(109)10 = (1101001)2
- B.
(123)5 = (212)4
- C.
(101)16 + (110)8 = (429)10
- D.
(241)6 = (131)8
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Correct answer: B
Concept: in a positional number system with base b, the digit positions carry the weights b0, b1, b2, b3 and so on, counting from the right. The value of a numeral is the sum of each digit multiplied by its own weight.
Two numerals written in different bases therefore denote the same number exactly when these weighted sums are equal, so a claim of the form (x)b = (y)c is settled by expanding both sides into base 10 and comparing the two totals.
Application: expanding each of the four statements gives the base-10 totals below.
Statement | Left side in base 10 | Right side in base 10 |
|---|---|---|
(109)10 = (1101001)2 | 109 | 64 + 32 + 8 + 1 = 105 |
(123)5 = (212)4 | 25 + 10 + 3 = 38 | 32 + 4 + 2 = 38 |
(101)16 + (110)8 = (429)10 | 257 + 72 = 329 | 429 |
(241)6 = (131)8 | 72 + 24 + 1 = 97 | 64 + 24 + 1 = 89 |
The two sides come out equal only for (123)5 = (212)4, where each side totals 38. In the other three statements the totals differ: 109 against 105, 329 against 429, and 97 against 89.
Cross-check: converting 38 back into each base by repeated division reproduces both numerals.
Dividing 38 by 5 repeatedly leaves the remainders 3, 2 and 1; read from the last remainder upwards this is (123)5.
Dividing 38 by 4 repeatedly leaves the remainders 2, 1 and 2; read from the last remainder upwards this is (212)4.
A common slip is to compare the digit strings instead of the values: (123)5 and (212)4 look nothing alike, yet both name the number 38.
Hence the statement that is true is (123)5 = (212)4.