Multiple choice

If (123) base 5 = (x3) y, then the number of possible values of x is

  1. 4

  2. 1

  3. 3

  4. 2
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D Correct answer
Explanation

(123)_5 = 1×25 + 2×5 + 3 = 38 in decimal. For (x3)_y = 38, we need x×y + 3 = 38, so x×y = 35. Factor pairs of 35 with x < y (required for valid digit in base y) are (1,35) and (5,7), giving x = 1 or 5. The base y must exceed the highest digit (3), so y = 35 or 7 are both valid.

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