AMC 10 · 2024 · #1
Grade 7 arithmeticalgebraPick an answer.
Brute-force multiplication of 9901 · 101 and 99 · 10101 would give two five-digit numbers and an easy place for a careless slip. Instead, notice that 9901 = 99 · 100 + 1 and 10101 = 101 · 100 + 1 — both numbers split into a product of the other given factors plus 1. That symmetry is exactly what Tool #16 looks for: rewrite the expression so a large piece cancels. Tool #4 (let a = 99, b = 101) makes the symmetry visible at a glance and turns the whole problem into one line of algebra.
Rewrite the big numbers
Both giants are a hundred times something, plus one.
Giving the repeated numbers short names is the Grade 6 "letters stand for numbers" move; it makes the hidden symmetry pop out.
6.EE.A.2Introduce A VariableSubstitute the letters
The whole expression becomes a formula in a and b.
Same expression, new clothing — the substitution doesn't change the value, but it lines up matching pieces.
6.EE.A.3Change Focus Count The ComplementExpand it
The same 100ab appears on both sides.
Distribute carefully, especially the minus sign across the second parenthesis.
A minus sign in front of a bracket reaches every term inside it, so the heavy parts cancel and only the small extras survive.
▸ Why?
A factor placed across a sum reaches every term inside it, and a minus sign is such a factor.
▸ Why?
A quantity added to its own opposite leaves nothing behind, so the matching heavy terms disappear.
Read the leftover
What is left is b minus a, namely 2.
Once the heavy term cancels, only the two tiny extras +b and -a survive — and their difference is just 2.
7.EE.A.1Change Focus Count The ComplementBig-looking numbers often hide a small structure. Naming 99 and 101 as a and b exposes the matching 100ab piece that cancels — leaving the AMC 12 opener as a Grade 7 distributive-property exercise!