For a nature number $$n=p_{1}^{q_{1}}\\times p_{2}^{q_{2}}\\times \\cdots \\times p_{m}^{q_{m}}$$ where $p_{i} (1 \\le i \\le m) $ are unique prime numbers.<\/p>\n\n\n\n
The total number of positive factors of $n$ is $$f=(q_{1}+1)\\times(q_{2}+1)\\times\\cdots\\times(q_{m}+1)$$ The sum of all positive factors of $n$ is $$\\sum_{d|n}d=\\dfrac{p_{1}^{q_1 +1}-1}{ p_1 -1}\\times \\dfrac{p_{2}^{q_{2}+1}-1}{ p_{2}-1}\\times \\cdots \\times\\dfrac{p_{m}^{q_{m}+1}-1}{p_{m} -1}$$ The product of all positive factors of $n$ is $$\\prod_{d|n}d=n^{\\frac{f}{2}}$$ i.e. the $\\dfrac{f}{2}th$ power of $n$, where $f$ is the total number of positive factors of $n$.<\/p>\n\n\n\n
Question 1<\/strong>: A number is perfect if the sum of all positive factors is twice of the number. What is the sum of the first $4$ perfect numbers found by Euclid?<\/p>\n\n\n\n Question 2<\/strong>: A natural number is multiplicative perfect if the square of the number is the product of all positive factors. What is the sum of the $3$ smallest multiplicative perfect numbers greater than $1$?<\/p>\n\n\n\n Question <\/strong>3: The sum of all positive factors of a natural number is 403. What is this number?<\/p>\n\n\n\n Question <\/strong>4: What is the largest natural number less than $1000$ such that the product of all positive factors is the number raised to the $12$th power?<\/p>\n","protected":false},"excerpt":{"rendered":" For a nature number $$n=p_{1}^{q_{1}}\\times p_{2}^{q_{2}}\\times \\cdots \\times p_{m}^{q_{m}}$$ where $p_{i} (1 \\le i \\le m) $ are unique prime numbers. The total number of positive factors of $n$ is $$f=(q_{1}+1)\\times(q_{2}+1)\\times\\cdots\\times(q_{m}+1)$$ The sum of all positive factors of $n$ is … Continue reading