AMC 8 · 2011 · #15
Grade 6 arithmeticProblem
How many digits are in the product 45⋅510?
Pick an answer.
AMC 8 2011 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: Find how many digits the integer $4^5 \cdot 5^{10}$ has when written out in base $10$.
Givens: Expression: $4^5 \cdot 5^{10}$; Bases $4$ and $5$ are different, so the exponents cannot be combined directly; Answer choices: (A) $8$, (B) $9$, (C) $10$, (D) $11$, (E) $12$
Unknowns: The number of digits in the product $4^5 \cdot 5^{10}$
Understand
Restated: Find how many digits the integer $4^5 \cdot 5^{10}$ has when written out in base $10$.
Givens: Expression: $4^5 \cdot 5^{10}$; Bases $4$ and $5$ are different, so the exponents cannot be combined directly; Answer choices: (A) $8$, (B) $9$, (C) $10$, (D) $11$, (E) $12$
Plan
Primary tool: #16 Change Focus / Change Your Point of View
Secondary: #9 Solve an Easier Related Problem
Computing $4^5 \cdot 5^{10}$ as a brute number is painful, but the bases $4$ and $5$ are hiding a $2$ inside $4$. Tool #16 (Change Focus) says: rewrite $4$ as $2^2$ so the bases become $2$ and $5$. Now the $2$s and $5$s pair up perfectly into $10$s, turning the product into a clean power of $10$. Tool #9 (Easier Related Problem) then takes over: counting digits in $10^{10}$ is far easier than in $4^5 \cdot 5^{10}$, and we already know the rule $10^n$ has $n+1$ digits.
Execute — Answer: D
6.EE.A.1 Step 1 - Rewrite $4$ using the smaller base $2$.
- Since $4 = 2^2$, we get $4^5 = (2^2)^5 = 2^{10}$ by the power-of-a-power rule.
💡 Changing the base from $4$ to $2$ lets the exponents talk to the $5^{10}$ next door.
6.EE.A.1 Step 2 - Substitute back into the original product.
- Now the two factors share the same exponent $10$, so we can combine them under a single power using $a^n \cdot b^n = (a \cdot b)^n$.
💡 Every $2$ finds a partner $5$ and together they form a $10$ — the whole product collapses into $10^{10}$.
5.NBT.A.2 Step 3 - Replace the original digit-counting question with an easier one: how many digits does $10^{10}$ have?
- Use the pattern $10^1 = 10$ ($2$ digits), $10^2 = 100$ ($3$ digits), $10^3 = 1000$ ($4$ digits).
- The rule is $10^n$ has $n+1$ digits — one $1$ followed by $n$ zeros.
💡 Multiplying by $10$ tacks on a zero — a Grade 5 place-value pattern.
5.NBT.A.2 Step 4 Apply the rule with $n = 10$: $10^{10}$ has $10 + 1 = 11$ digits.
💡 One leading $1$ plus ten trailing zeros equals eleven digits total.
6.EE.A.1 Rewrite $4$ using the smaller base $2$. Since $4 = 2^2$, we get $4^5 = (2^2)^5 = 6.EE.A.1 Substitute back into the original product. Now the two factors share the same ex 5.NBT.A.2 Replace the original digit-counting question with an easier one: how many digits 5.NBT.A.2 Apply the rule with $n = 10$: $10^{10}$ has $10 + 1 = 11$ digits. Review
Reasonableness: Quick sanity check on size: $4^5 = 1024$ ($4$ digits) and $5^{10} = 9{,}765{,}625$ ($7$ digits). Multiplying a $4$-digit number by a $7$-digit number gives a result with either $4 + 7 - 1 = 10$ or $4 + 7 = 11$ digits. Our answer $11$ lands in that range, and since the leading digits $1.024 \times 9.765\ldots \approx 10$ push the product over $10^{10}$, the upper case ($11$ digits) is correct. Choices (A)$8$ and (B)$9$ are far too small; (E)$12$ would require the product to exceed $10^{11}$, which it does not.
Alternative: Tool #5 (Look for a Pattern) on digit counts: $10^1$ has $2$ digits, $10^2$ has $3$, ..., $10^n$ has $n+1$. Once we've rewritten the product as $10^{10}$, the pattern instantly delivers $10 + 1 = 11$. Or, without rewriting, multiply $4^5 \cdot 5^{10} = 1024 \cdot 9{,}765{,}625 = 10{,}000{,}000{,}000$ directly and count $11$ digits — but recognizing the power of $10$ first avoids the long multiplication entirely.
CCSS standards used (min grade 6)
6.EE.A.1Write and evaluate numerical expressions involving whole-number exponents (Rewriting $4^5 = (2^2)^5 = 2^{10}$ and combining $2^{10} \cdot 5^{10} = (2 \cdot 5)^{10} = 10^{10}$ using the exponent rules.)5.NBT.A.2Explain patterns in the number of zeros of the product when multiplying by powers of 10 (Recognizing that $10^{10}$ is written as one $1$ followed by ten zeros, giving $10 + 1 = 11$ digits.)
⭐ Whenever you see $2$s and $5$s with matching exponents, pair them into $10$s — the answer just falls out as a power of $10$.
⭐ Whenever you see $2$s and $5$s with matching exponents, pair them into $10$s — the answer just falls out as a power of $10$.
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