Consider the equation \((43)_x\) = \((y3)_8\) where 𝑥 and 𝑦 are unknown. The…
2015
Consider the equation \((43)_x\) = \((y3)_8\) where 𝑥 and 𝑦 are unknown. The number of possible solutions is ______________
Attempted by 510 students.
Show answer & explanation
Correct answer: 5
Key idea: convert both representations to base 10 and equate them.
Express the numbers in base 10: (43)_x = 4x + 3 and (y3)_8 = 8y + 3.
Equate them: 4x + 3 = 8y + 3 ⇒ 4x = 8y ⇒ x = 2y.
Apply digit/base constraints: For (43)_x the base x must be at least 5 (since digit 4 appears). For (y3)_8, y is a base-8 digit and as a leading digit must be 1 through 7.
Solve x = 2y with y = 1..7 and require x ≥ 5: y = 1 → x = 2 (invalid); y = 2 → x = 4 (invalid); y = 3 → x = 6 (valid); y = 4 → x = 8 (valid); y = 5 → x = 10 (valid); y = 6 → x = 12 (valid); y = 7 → x = 14 (valid).
Thus there are five valid (x,y) pairs: (6,3), (8,4), (10,5), (12,6), (14,7).
Answer: 5