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Q.
Mean value of odd divisors of $3630$, which leaves the remainder $1$, when divided by $4$ is
Statistics
Solution:
$3630 =2^{1}\,3^{1}\,5^{1}\,11^{2}=2^{a}\,3^{b}\,5^{c}\,11^{d}$
$\therefore $ Number of odd divisors of $3630$ are $(b+1) (c+1) (d+1) =2. 2. 3$
$\therefore $ Total number of odd divisors of $3630 = 12$
These divisors are the factors of $(3 \times 5\times 11^{2})=1815$
which are $1 ,3 , 5, 11, 15, 33, 55, 121, 165,363, 605, 1815$ From these divisors w e need those divisor w hich leaves the remainder 1, when divided by 4
$\therefore $ Those divisors are $1, 5, 33, 121, 165, 605$
$\therefore $ Number of such divisors $= 6$
$\therefore $ Mean of such divisors =
$\frac{\text{ Sum of all odd divisors which leave remainder 1
when divided by 4}}{\text{ Number of divisors of the form} (4k +1)}$
$=\frac{1+5+33+121+165+605}{6}$
$=\frac{930}{6}=155$