When you only know the quotient
Number Theory
Level
pending
The division between two natural numbers has a quotient of 16 and leaves the highest remainder possible. The sum of the dividend and the divisor is 125. Find the remainder.
The answer is 6.
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1 solution
Let the dividend be n and the divisor m . Then n + m = 1 2 5 ⟹ n = 1 2 5 − m . Upon division the highest possible remainder will be m − 1 , so we are looking for m , n such that
m n = 1 6 + m m − 1 ⟹ n = 1 6 m + ( m − 1 ) = 1 7 m − 1 .
Substituting n = 1 2 5 − m then gives us that
1 2 5 − m = 1 7 m − 1 ⟹ 1 2 6 = 1 8 m ⟹ m = 7 ,
and so the desired remainder is m − 1 = 6 .