AMC 8 · 2024 · #1

Grade 4 arithmetic

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

📏 Medium solution 💡 3 insights

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Problem

What is the unit digit of: $222{,}222-22{,}222-2{,}222-222-22-2?$
$\textbf{(A) } 0\qquad\textbf{(B) } 2\qquad\textbf{(C) } 4\qquad\textbf{(D) } 8\qquad\textbf{(E) } 10$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: An expression with one large number minus five smaller numbers: $222{,}222 - 22{,}222 - 2{,}222 - 222 - 22 - 2$; Every number in the expression is written using only the digit 2; Every number ends in 2 (units digit is 2 for every term); Five answer choices: (A) 0, (B) 2, (C) 4, (D) 8, (E) 10

Unknowns: The units digit of the final result

Understand

Restated: 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.

Plan

Primary tool: #9 Solve an Easier Related Problem

Secondary: #5 Look for a Pattern, #3 Eliminate Possibilities

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.

Execute — Answer: B

#9 Solve an Easier Related Problem 4.NBT.A.2 Step 1

Notice that every number in the expression — $222{,}222$, $22{,}222$, $2{,}222$, $222$, $22$, and $2$ — ends in the digit 2.
Because the units digit of any sum or difference depends only on the units digits of the numbers involved, we can ignore every digit except the rightmost one.
This is the "easier related problem" move: we shrink each six-digit (or smaller) number down to its units digit and work with single digits.

$$222{,}222 \to 2,\ 22{,}222 \to 2,\ 2{,}222 \to 2,\ 222 \to 2,\ 22 \to 2,\ 2 \to 2$$

💡 Reading a multi-digit number and naming its units digit is exactly the place-value skill from Grade 4.

#5 Look for a Pattern 2.OA.B.2 Step 2

Group the five numbers being subtracted and add their units digits.
Since each of the five contributes a 2 in the ones column, the sum of those units digits is $2+2+2+2+2 = 10$.
So the five subtracted numbers together end in the digit 0 (the units digit of 10).

$$2 + 2 + 2 + 2 + 2 = 10 \;\Rightarrow\; \text{units digit} = 0$$

💡 Adding five small numbers within 20 is fluent mental math from Grade 2.

#9 Solve an Easier Related Problem 4.NBT.B.4 Step 3

Now the original expression behaves, in its ones column, like $2 - 0 = 2$.
The first number ($222{,}222$) ends in 2, and the combined thing we are subtracting ends in 0, so no borrowing happens in the ones column.
The units digit of the final answer is therefore 2.

$$\underbrace{222{,}222}_{\text{ends in }2} - \underbrace{(22{,}222 + 2{,}222 + 222 + 22 + 2)}_{\text{ends in }0} \;\Rightarrow\; 2 - 0 = 2$$

💡 Subtracting multi-digit numbers column-by-column (Grade 4) lets us isolate the ones column safely.

#3 Eliminate Possibilities 2.NBT.A.1 Step 4

Cross-check against the answer choices.
The candidates are 0, 2, 4, 8, 10.
Our units digit is 2, which matches choice (B).
Choice (E) 10 is not even a single digit, so it can be eliminated on form alone.
The other digits 0, 4, 8 don't arise from $2 - 0$.

$$\text{Units digit} = 2 \;\Rightarrow\; \textbf{(B)}$$

💡 Identifying the ones place of a number to match it to a multiple-choice option uses place value from Grade 2.

[1] #9 4.NBT.A.2 Notice that every number in the expression — $222{,}222$, $22{,}222$, $2{,}222$,

[2] #5 2.OA.B.2 Group the five numbers being subtracted and add their units digits. Since each o

[3] #9 4.NBT.B.4 Now the original expression behaves, in its ones column, like $2 - 0 = 2$. The f

[4] #3 2.NBT.A.1 Cross-check against the answer choices. The candidates are 0, 2, 4, 8, 10. Our u

Review

Reasonableness: 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.

Alternative: An alternative is Tool #13 (Convert to Algebra) using modular arithmetic: compute the entire expression $\bmod\ 10$. Each term is $\equiv 2 \pmod{10}$, so the result is $2 - 2 - 2 - 2 - 2 - 2 = -8 \equiv 2 \pmod{10}$. Same answer, but the easier-problem approach is more natural for an elementary student.

CCSS standards used (min grade 4)

2.NBT.A.1 Understand that the three digits of a three-digit number represent hundreds, tens, and ones (Identifying the ones place of each number so we can compare against the answer choices.)
2.OA.B.2 Fluently add and subtract within 20 using mental strategies (Adding the five units digits $2+2+2+2+2 = 10$ mentally.)
4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Recognizing that each multi-digit number (up to $222{,}222$) has units digit 2.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Reasoning column-by-column about the multi-digit subtraction so the units digit can be read directly from the ones column.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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