2013 AIME I Problems/Problem 2
Revision as of 02:30, 3 October 2020 by Pleaseletmewin (talk  contribs)
Contents
Problem 2
Find the number of fivedigit positive integers, , that satisfy the following conditions:

(a) the number is divisible by

(b) the first and last digits of are equal, and

(c) the sum of the digits of is divisible by
Solution
The number takes a form of , in which . Let and be arbitrary digits. For each pair of , there are exactly two values of that satisfy the condition of . Therefore, the answer is
This 9line code in Python also gives the answer too. import math counter=0 for integer in range(10000,99999):
if str(integer)[4] != '5' or str(integer)[0] != '5': counter=counter else: if math.remainder(int(str(integer)[1]+str(integer)[2]+str(integer)[3]),5)==0: counter+=1
print(counter)
Video Solution
https://www.youtube.com/watch?v=kz3ZX4PT_0 ~Shreyas S
See also
2013 AIME I (Problems • Answer Key • Resources)  
Preceded by Problem 1 
Followed by Problem 3  
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15  
All AIME Problems and Solutions 
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.