AMC 8 · 2024 · #1

Grade 4 arithmetic

Archetype Units-Digit Tracking └ Sub-archetype Digitwise Additive Reduction

units-digit-trackingplace-valuemental-arithmetic units-digit-trackingmodular-arithmetic-mod-10 ↑ Prerequisites: multi-digit-arithmeticplace-value

📏 Medium solution 💡 3 insights

📘 View easy version →

Problem

We are asked for the units (ones) digit of the value of 222,222 - 22,222 - 2,222 - 222 - 22 - 2. We do NOT need the full value, only the last digit.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2024 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

The numbers look scary (six digits!), but because we only need the last digit, we can replace each big number with just its units digit. That turns the whole problem into single-digit arithmetic — a much easier related problem (Tool #9). The pattern of "every number ends in 2" (Tool #5) makes the shortcut leap out, and since this is multiple-choice, Tool #3 lets us verify by elimination at the end.

1STEP 1

Every number here — 222,222 down to 2 — ends in the digit 2, so keep only each ones digit and drop the rest.

222,222 → 2, 22,222 → 2, 2,222 → 2, 222 → 2, 22 → 2, 2 → 2

2STEP 2

Add the five subtracted ones digits: 2+2+2+2+2 = 10, so together they end in 0.

2 + 2 + 2 + 2 + 2 = 10 → units digit = 0

3STEP 3

In the ones column the whole thing acts like 2 - 0 = 2; the leading 2 stays and no borrow occurs.

222,222_ends in 2 - (22,222 + 2,222 + 222 + 22 + 2)_ends in 0 → 2 - 0 = 2

4STEP 4

Match against the choices: our ones digit 2 is choice (B); 10 isn't a single digit, and 0, 4, 8 can't come from 2 - 0.

Units digit = 2 → (B)

Answer

Quick sanity check: the actual value of the expression is 222,222 - 24,702 = 197,520. Wait — let me re-add: 22,222 + 2,222 + 222 + 22 + 2 = 24,690, and 222,222 - 24,690 = 197,532. The last digit of 197,532 is indeed 2. The units-digit shortcut agrees with the full subtraction, so the answer (B) 2 is correct.

💡Key takeaway

This AMC 8 problem only needs Grade 4 multi-digit place-value sense you already know!

More like this