Ratio: Sum of Odd Factors to Sum of Even Factors
October 31, 2025 2025-10-31 15:29Ratio: Sum of Odd Factors to Sum of Even Factors
Ratio of Sum of Odd Factors to Sum of Even Factors
Let \(N=2^a\cdot M\) with \(M\) odd and \(a\ge 1\). Then \[ \frac{\text{sum of odd factors of }N}{\text{sum of even factors of }N} = \frac{1}{\,2+2^2+\cdots+2^a\,} = \frac{1}{2(2^a-1)}. \] For odd \(N\) (\(a=0\)), there are no even factors, so the ratio is undefined/infinite.
Why this works (one-line proof)
Worked examples (full questions)
Example 1
Question. Let \(N=360\). Find the ratio \(\displaystyle \frac{\text{sum of odd factors of }N}{\text{sum of even factors of }N}\).
Solution
\(360=2^3\cdot 3^2\cdot 5\Rightarrow a=3,\; M=3^2\cdot 5\). Ratio \(=\dfrac{1}{2+4+8}=\dfrac{1}{14}\). (Check: \(\sigma(M)=(1+3+9)(1+5)=78\); \(\sigma(2^3)=15\); \(\sigma(N)=1170\); even-sum \(=1170-78=1092\); \(78/1092=1/14\).)Example 2
Question. Let \(N=48\). Compute \(\displaystyle \frac{\text{sum of odd divisors of }N}{\text{sum of even divisors of }N}\).
Solution
\(48=2^4\cdot 3\Rightarrow a=4\). Ratio \(=\dfrac{1}{2+4+8+16}=\dfrac{1}{30}\).Example 3
Question. For \(N=2^a\) with \(a\ge 1\), express \(\displaystyle \frac{\text{sum of odd factors of }N}{\text{sum of even factors of }N}\) in terms of \(a\). Also evaluate it for \(a=5\).
Solution
Here \(M=1\) (odd), so ratio \(=\dfrac{1}{2+2^2+\cdots+2^a}=\dfrac{1}{2(2^a-1)}\). For \(a=5\): ratio \(=\dfrac{1}{2(32-1)}=\dfrac{1}{62}\).Example 4
Question. Let \(N=750\). Find the ratio of sum of odd factors of \(N\), to the sum of even factors of \(N\).
Solution
\(750=2^1\cdot 3^1\cdot 5^3\Rightarrow a=1,\; M=3\cdot 5^3\). Odd-sum \(=\sigma(3)\sigma(5^3)=(1+3)(1+5+25+125)=4\cdot156=624\). Even-sum \(=(2+2^2+\cdots+2^1)\cdot\text{odd-sum}=2\cdot 624=1248\). Ratio \(=624/1248=1/2\).Example 5 (edge case)
Question. If \(N\) is odd (e.g., \(N=945\)), what is \(\displaystyle \frac{\text{sum of odd factors}}{\text{sum of even factors}}\)? Explain.
Solution
For odd \(N\) we have \(a=0\). There are no even divisors, so the denominator is \(0\); the ratio is undefined (sometimes informally called “infinite”). \(\sigma(N)\) itself is finite; only the even-part sum is \(0\).Quick practice
- \(N=2^5\cdot 45\). Compute the ratio. (Ans: \(1/(2+4+8+16+32)=1/62\))
- \(N=750=2\cdot 3\cdot 5^3\). Find the sum of odd factors and the sum of even factors. (Hint: odd sum \(=\sigma(3)\sigma(5^3)\); even \(=\) odd sum \(\times (2+2^2+\cdots+2^1)=\) odd sum \(\times 2\)).
- If \(N\) is odd, explain why the ratio is undefined. What is \(\sigma(N)\) in this case?
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