Problem:
Let N be the unique positive integer such that dividing 273436 by N leaves a remainder of 16 and dividing 272760 by N leaves a remainder of 15. What is the tens digit of N?
Answer Choices:
A. 0
B. 1
C. 2
D. 3
E. 4
Solution:
It is given that dividing 273436 by N leaves a remainder of 16, and dividing 272760 by N leaves a remainder of 15.
N∣(273436−16)=273420,N∣(272760−15)=272745.
so N∣(273420−272745)=675. Thus N divides both 273420 and 675, or N∣gcd(273420,675).
Since gcd(273420,675)=45, therefore, N∣45. The possible divisors of 45 are 1,3,5,9,15,45. Since the remainder 16<N, therefore, the only possible value of N=45.
The tens digit of N is
(E) 4.
The problems on this page are the property of the MAA's American Mathematics Competitions