AMC 10 · 2010 · #6
Grade 4 number-theoryPick an answer.
The key pressure is the size jump: adding only 32 pushes a three-digit number up to four digits, so x must sit right at the top of the three-digit range. Pinning down that boundary shrinks the search to just a handful of large palindromes, and then each one can be tested directly by adding 32 and checking whether the result reads the same both ways.
Force x near the top
Gaining a digit forces the number near the top.
A small +32 can only cross into four digits if x is already almost 1000.
A small addition can only push a number into four digits if it already sits almost at a thousand.
▸ Why?
The number of digits is set by which power-of-ten window the number sits in.
▸ Why?
Adding a small amount can carry a number across at most one such boundary, so the range is tightly bounded.
List the palindromes there
Only four palindromes live there.
In this range only numbers whose first and last digit agree are palindromes.
4.NBT.A.2Eliminate PossibilitiesAdd 32 and check each
Testing each finds exactly one that works.
Just try the few survivors; only one lands on a symmetric four-digit number.
4.NBT.B.4Guess And CheckAdd the digits of x
Adding its digits gives 24, choice (E).
The question wants the digit sum, so just add the three digits of the winner.
4.NBT.B.4Guess And CheckWhen a small addition forces a number into more digits, the starting number must be sitting right at the edge, so look there first.
- Force x near the top
- List the palindromes there
- Add 32 and check each
- Add the digits of x