Question

Find the number of factors and sum of factors of 25920 excluding 1 and itself

Answer 1

No. of factors of a number can be found by the following steps:

⇒  Do prime factorization of the number and write the number as product of prime numbers in exponential form.

Like, 12 = 2² × 3

⇒  Add 1 to each exponents of the prime factors.

⇒ Then multiply the numbers thus obtained.

Okay, considering 25920,

$\begin{tabular}{c|l}2&25920\\ \cline{2-}2&12960\\ \cline{2-}2&6480\\ \cline{2-}2&3240\\ \cline{2-}2&1620\\ \cline{2-}2&810\\ \cline{2-}5&405\\ \cline{2-}3&81\\ \cline{2-}3&27\\ \cline{2-}3&9\\ \cline{2-}&3\\ \cline{2-}\end{tabular}$

Thus, 25920 = 2⁶ × 3⁴ × 5¹

Now, we have to add 1 to each exponents.

Exponent of 2 is 6   →   6 + 1 = 7

Exponent of 3 is 4    →   4 + 1 = 5

Exponent of 5 is 1    →    1 + 1 = 2

Now, multiply these numbers...

7 × 5 × 2 = 70

So there are a total of 70 factors of 25920, including 1 and itself. Thus there are 68 factors excluding them.

Now, we have to find the sum of these 68 factors. For this, we have to understand the given concept below.

$\textsf{Let N be an integer. If the prime factorization of N is taken as \ p^x \times q^y \times r^z ,} \\ \\ \textsf{then the sum of all factors of N will be,} \\ \\ \\ (1+p+p^2+...+p^x)(1+q+q^2+...+q^y)(1+r+r^2+...+r^z)$

So, considering  25920 = 2⁶ × 3⁴ × 5¹,

the sum of all factors of 25920 will be,

$(1+2+2^2+2^3+2^4+2^5+2^6)(1+3+3^2+3^3+3^4)(1+5) \\ \\ \\ (1+2+4+8+16+32+64)(1+3+9+27+81)(1+5) \\ \\ \\ 127 \times 121 \times 6 = \large \text{92202}$

This 92202 is the sum of all 70 factors of 25920. To get the sum of the factors excluding 1 and 25920, we have to deduct 1 and 25920 from this sum.

So,

$92202-(1+25920)=\Large \textbf{66281}$

Thus 66281 is the sum of the 68 factors.

Answer 2

Answer:

25920= 26 x 34 x5¹

No. of factors (6+1) (4+1)(1+1)=70

. No. of factors excluding 1 and itself = 68

and Sum of factors = [2^(6+1)-1] /2-1× [3^(4+1)-1]/3-1 ×[5^(1+1)-1] /5-1

=127x121×6=92202

.. Sum of factors excluding 1 and itself=92202-(1+25920) = 66281

Step-by-step explanation:

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