Finding sum of factors of a number using prime factorization

Your approach works fine: $2450=2\cdot 5^2\cdot 7^2$, therefore the sum of divisor is $$(1+2)(1+5+25)(1+7+49)=5301=3\cdot 1767.$$

You are looking for the Formula For Sum Of Divisors, from there:

Each of these sums is a geometric series; hence we may use the formula for sum of a geometric series to conclude $$ \sum_{d|n}d = \prod_{i=1}^k \frac{p_i^{m_i+1}-1}{p_i-1} $$


Solving for $1225 = 5^2 × 7^2$

Sum of divisors = $(a^{p+1} – 1)/(a – 1) × (b^{q+1} – 1)/(b – 1)$
Here $a = 5$, $b = 7$ prime factors
$p = 2$ and $q = 2$
Sum of divisors = $(5^3 – 1)/(5 – 1) × (7^3 – 1)/(7 – 1)$
= $(124/4) × (342/6)$
= $(31 × 57)$
= $1767$


$2450=2\cdot5^2\cdot7^2$, so the algorithm gives

$$(1+2)(1+5+25)(1+7+49)=3\cdot31\cdot57=5301\;.$$

The divisors of $2450$ are $1,2,5,7,10,14,25,35,49,50,70,98,175,245,350,490,1225$, and $2450$, whose sum is indeed $5301$.

It’s not difficult to prove that the algorithm works, so if you thought that it failed, you made a mistake somewhere in your calculations; my guess would be that you missed a divisor of $2450$.

Tags:

Elementary Number Theory

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