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If \(n\) is a positive integer such that \(2^{51}\) and \((2^n)(3^3)\) have the same number of positive factors, what is the value of \(n\)?
Let's start by understanding what we're looking for in everyday terms. We have two numbers: \(2^{51}\) and \((2^n)(3^3)\). The problem tells us these two numbers have the same number of positive factors, and we need to find what value of n makes this true.
Think of positive factors as all the whole numbers that divide evenly into our number. For example, the number 12 has factors 1, 2, 3, 4, 6, and 12 - that's 6 factors total.
So our job is to count how many factors each expression has and set them equal.
Process Skill: TRANSLATE
There's a beautiful pattern for counting factors when we know the prime factorization of a number. Let me explain this with a simple example first.
Consider \(2^3 = 8\). The factors are: \(2^0 = 1\), \(2^1 = 2\), \(2^2 = 4\), \(2^3 = 8\). That's 4 factors total. Notice that we can choose the exponent of 2 to be 0, 1, 2, or 3 - that's 4 choices, which gives us 4 factors.
For \(2^3 \times 3^2 = 8 \times 9 = 72\), we can choose the exponent of 2 to be 0, 1, 2, or 3 (4 choices) AND independently choose the exponent of 3 to be 0, 1, or 2 (3 choices). This gives us \(4 \times 3 = 12\) total factors.
The general rule: If a number has prime factorization \(p_1^{a_1} \times p_2^{a_2} \times \ldots \times p_k^{a_k}\), then the number of positive factors is \((a_1 + 1)(a_2 + 1)\ldots(a_k + 1)\).
For \(2^{51}\), we have just one prime (2) raised to the power 51.
Using our formula: The number of factors = \(51 + 1 = 52\).
This means \(2^{51}\) has exactly 52 positive factors.
Now let's look at \((2^n)(3^3)\). This has:
- The prime 2 raised to the power n
- The prime 3 raised to the power 3
Using our formula: The number of factors = \((n + 1)(3 + 1) = (n + 1)(4) = 4(n + 1)\).
Since both expressions must have the same number of factors:
\(4(n + 1) = 52\)
Now we solve the equation:
\(4(n + 1) = 52\)
Divide both sides by 4:
\(n + 1 = 13\)
Subtract 1 from both sides:
\(n = 12\)
Let's verify: If \(n = 12\), then \((2^{12})(3^3)\) has \((12 + 1)(3 + 1) = 13 \times 4 = 52\) factors, which matches the 52 factors of \(2^{51}\). ✓
The value of n is 12.
Looking at our answer choices, this corresponds to choice A.
1. Confusing "number of factors" with the actual factors themselves
Students often get distracted by trying to list out all the actual factors (like 1, 2, 4, 8, 16...) instead of just counting them. This leads to unnecessary work and potential errors, especially with large numbers like \(2^{51}\).
2. Forgetting or misremembering the factor counting formula
Many students either don't recall that for \(p_1^{a_1} \times p_2^{a_2} \times \ldots \times p_k^{a_k}\), the number of factors is \((a_1 + 1)(a_2 + 1)\ldots(a_k + 1)\), or they remember it incorrectly as just \(a_1 \times a_2 \times \ldots \times a_k\) (forgetting to add 1 to each exponent).
1. Arithmetic errors when applying the factor formula
Students might correctly identify that \(2^{51}\) has \((51 + 1) = 52\) factors, but then make errors when calculating \((n + 1)(3 + 1) = 4(n + 1)\), perhaps writing it as \((n + 1)(4 + 1) = 5(n + 1)\) instead.
2. Algebraic mistakes when solving 4(n + 1) = 52
Common errors include dividing incorrectly (getting \(n + 1 = 52/4 = 13\) but then forgetting to subtract 1), or making basic arithmetic mistakes like \(52 \div 4 = 14\) instead of 13.
3. Errors while selecting the answer
No likely faltering points - once students correctly solve \(n + 1 = 13\) to get \(n = 12\), the answer choice A is straightforward to identify.