AMC 10 · 2021 · #10

Grade 9 number-theory

Archetype Small-Domain Constrained Search └ Sub-archetype Factor-Pair Enumeration

simons-favorite-factoring-tricktriangular-numbersprime-factorizationfactors convert-to-algebrasystematic-enumerationbound-inequality-then-enumerate ↑ Prerequisites: factorstriangular-numbers

📏 Long solution 💡 3 insights

Problem

Pick two different numbers from 1 through 37 and take them out. The thirty-five numbers left behind add up to exactly the product of the two you took out. Find how far apart those two numbers are.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

How to solve

Strategy Introduce a Variable

The two selected numbers are the only unknowns, and the whole problem is one sentence linking them. Naming them turns that sentence into a single equation that contains both their product and their sum. One equation with two unknowns normally has endless solutions, so the equation alone is not enough. The escape route is to rewrite it as one product of two brackets equal to a fixed number. Once it has that shape, the fact that both numbers are whole and bounded does the rest of the work, because a fixed number has only a handful of factor pairs.

1STEP 1

Total the whole set

Add up the whole range.

1 + 2 + … + 37 = (37 · 38)/2 = 703

2STEP 2

Name the two numbers

The condition becomes one equation.

703 - a - b = ab ⟹ ab + a + b = 703

3STEP 3

Add one to force a factorization

Adding one lets it factor.

ab + a + b + 1 = 704 ⟹ (a+1)(b+1) = 704

4STEP 4

List the factor pairs

List the factor pairs of the right side.

704 = 2⁶ · 11: 1 · 704, 2 · 352, 4 · 176, 8 · 88, 11 · 64, 16 · 44, 22 · 32

5STEP 5

Use the range to keep one pair

The range leaves only one pair.

2 ≤ a+1 ＜ b+1 ≤ 38 ⟹ {a+1, b+1} = {22, 32}

6STEP 6

Undo and check

The difference is 10.

a = 21, b = 31; 21 · 31 = 651 = 703 - 21 - 31; 31 - 21 = 10

Answer

The factoring argument leaves exactly one candidate pair, and that pair passes the original condition without rounding: 21 times 31 equals 651, and the 35 leftover numbers total 703 minus 52, which is also 651. A rough size check agrees too. The product has to land near 703, so the two numbers should multiply to roughly 700, and 21 times 31 equals 651 with their sum 52 closing the gap exactly. Neighbouring guesses fail: 20 and 33 give a product of 660 while the leftovers total 703 minus 53, which is 650. Both numbers stay inside 1 to 37 and are distinct, and the difference 10 is one of the five offered choices.

💡Key takeaway

When an equation mixes the product and the sum of two unknowns, add the one term that completes the factorization, and the problem turns into listing factor pairs.

Total the whole set first
Name the two numbers
Add 1 to force a factorization
List every factor pair of 704
Use the range to kill all but one pair
Undo the shift and check