AMC 8 · 2012 · #12
Grade 4 number-theoryProblem
What is the units digit of ?
Pick an answer.
Toolkit + CCSS Solution
Understand
Restated: What single digit appears in the ones place of $13^{2012}$? We are not asked for the whole number, only its last digit.
Givens: Base = $13$; Exponent = $2012$; Answer choices: (A) $1$, (B) $3$, (C) $5$, (D) $7$, (E) $9$
Unknowns: The units (ones) digit of $13^{2012}$
Understand
Restated: What single digit appears in the ones place of $13^{2012}$? We are not asked for the whole number, only its last digit.
Givens: Base = $13$; Exponent = $2012$; Answer choices: (A) $1$, (B) $3$, (C) $5$, (D) $7$, (E) $9$
Plan
Primary tool: #5 Look for a Pattern
Secondary: #9 Solve an Easier Related Problem
The exponent $2012$ is way too big to compute, so we use Tool #9 (Easier Problem): compute the first few powers $3^1, 3^2, 3^3, \dots$ and look only at their ones digits. Tool #5 (Look for a Pattern) then takes over — powers of $3$ cycle through ones digits $\{3, 9, 7, 1\}$ every $4$ steps. Once the cycle length is known, finding the $2012$th term becomes a single division-with-remainder problem. We avoid Tool #13 (Algebra) because a clean cycle pattern makes algebra unnecessary.
Execute — Answer: A
4.NBT.B.5 Step 1 - Strip the problem down.
- The ones digit of a product depends only on the ones digits of its factors, so the ones digit of $13 \times 13 \times \dots$ matches the ones digit of $3 \times 3 \times \dots$.
- Replace $13^{2012}$ with the easier $3^{2012}$.
💡 Multiplying multi-digit numbers and tracking how the ones place comes out of $(\text{tens} + \text{ones}) \times (\text{tens} + \text{ones})$ is a Grade 4 place-value insight.
4.NBT.B.5 Step 2 Build a small table of powers of $3$ and record only the ones digit of each.
💡 Computing $3 \times 3, 9 \times 3, 27 \times 3, \dots$ is straight Grade 4 multi-digit multiplication.
4.OA.C.5 Step 3 - Spot the repeating pattern.
- The ones digits read $3, 9, 7, 1, 3, 9, 7, 1, \dots$ — a block of four that repeats.
- The cycle is $\{3, 9, 7, 1\}$ with length $4$.
💡 Generating a sequence from a rule and noticing it repeats is exactly the Grade 4 "analyze patterns" standard.
4.NBT.B.6 Step 4 - Use the cycle to jump to the $2012$th term.
- Divide the exponent by the cycle length and look at the remainder: positions $1, 2, 3, 0$ in the cycle map to digits $3, 9, 7, 1$.
💡 Finding a whole-number quotient with remainder for a $4$-digit number divided by a $1$-digit divisor is the Grade 4 division standard.
4.OA.C.5 Step 5 - Read the result.
- Remainder $0$ means $3^{2012}$ lands on the same spot in the cycle as $3^4$ — the last entry of the block, whose ones digit is $1$.
- So the ones digit of $13^{2012}$ is $1$.
💡 Mapping a position in a repeating cycle back to its value is the same pattern-analysis skill the table is built on.
4.NBT.B.5 Strip the problem down. The ones digit of a product depends only on the ones dig 4.NBT.B.5 Build a small table of powers of $3$ and record only the ones digit of each. 4.OA.C.5 Spot the repeating pattern. The ones digits read $3, 9, 7, 1, 3, 9, 7, 1, \dots$ 4.NBT.B.6 Use the cycle to jump to the $2012$th term. Divide the exponent by the cycle len 4.OA.C.5 Read the result. Remainder $0$ means $3^{2012}$ lands on the same spot in the cy Review
Reasonableness: Sanity-check the cycle on an exponent we can verify: $3^8 = 6561$, which ends in $1$. Our rule says $8 \div 4$ has remainder $0$, so position "last in cycle" $= 1$. Matches. Try $3^{10} = 59049$, ending in $9$: $10 \div 4 = 2$ remainder $2$, so position $2$ in the cycle $= 9$. Matches again. The cycle is reliable, so $3^{2012}$ ending in $1$ is trustworthy.
Alternative: Tool #2 (Systematic List) could replace Tool #9 here: list ones digits of $3^1, 3^2, \dots, 3^8$ in order until the repetition is obvious, then count by groups of $4$. Tool #3 (Eliminate) is weaker on this problem — no choice can be ruled out just by inspection, since all five digits $\{1, 3, 5, 7, 9\}$ are plausible ones digits of an odd power.
CCSS standards used (min grade 4)
4.NBT.B.5Multiply a whole number of up to four digits by a one-digit whole number (Reducing $13^{2012}$ to $3^{2012}$ by place-value reasoning, and computing the small powers $3^1, 3^2, 3^3, 3^4, 3^5$ to expose the ones-digit pattern.)4.OA.C.5Generate a number or shape pattern that follows a given rule and identify apparent features (Recognizing that the ones digits of powers of $3$ form the repeating cycle $(3, 9, 7, 1)$ and using that cycle to predict the $2012$th term.)4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors (Computing $2012 \div 4 = 503$ remainder $0$ to locate position $2012$ inside the length-$4$ cycle.)
⭐ This AMC 8 problem only needs Grade 4 pattern-spotting — find the cycle, divide to find the position, and read off the digit!
⭐ This AMC 8 problem only needs Grade 4 pattern-spotting — find the cycle, divide to find the position, and read off the digit!
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