Maths Olympiad Prep

Track / Stage 3 / 132 of 260 #132 of 1964

Problem 132

AMC 10/12, early questions
Number theory Difficulty 3.5 Multiple choice

The hundreds digit of a three-digit number is 22 more than the units digit. The digits of the three-digit number are reversed, and the result is subtracted from the original three-digit number. What is the units digit of the result?

Pick one

Official solution

Let the hundreds, tens, and units digits of the original three-digit number be aa, bb, and cc, respectively. We are given that a=c+2a=c+2. The original three-digit number is equal to 100a+10b+c=100(c+2)+10b+c=101c+10b+200100a+10b+c = 100(c+2)+10b+c = 101c+10b+200. The hundreds, tens, and units digits of the reversed three-digit number are cc, bb, and aa, respectively. This number is equal to 100c+10b+a=100c+10b+(c+2)=101c+10b+2100c+10b+a = 100c+10b+(c+2) = 101c+10b+2. Subtracting this expression from the expression for the original number, we get (101c+10b+200)(101c+10b+2)=198(101c+10b+200) - (101c+10b+2) = 198. Thus, the units digit in the final result is (E) 8\boxed{\textbf{(E)}\ 8}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.