GIVEN:

N = 4200

CONCEPT:

Let there be a composite number N and its prime factors be a, b, c, d, … etc. and p, q, r, s, … etc. be the indices (or powers) of the a, b, c, d, … etc respectively i.e., if N can be expressed as –

N = a^p. b^q.c^r.d^s…..

∴ The sum of the factors of N is \(\frac{{\left( {{{\rm{a}}^{{\rm{p\;}} + {\rm{\;}}1}} - 1} \right)\left( {{{\rm{b}}^{{\rm{q\;}} + {\rm{\;}}1}} - 1} \right)\left( {{{\rm{c}}^{{\rm{r\;}} + {\rm{\;}}1}} - 1} \right)\left( {{{\rm{d}}^{{\rm{s\;}} + {\rm{\;}}1}} - 1} \right)}}{{\left( {{\rm{a}} - 1} \right)\left( {{\rm{b}} - 1} \right)\left( {{\rm{c}} - 1} \right)\left( {{\rm{d}} - 1} \right)}}\)

CALCULATION:

The factor of 4200 is-

⇒ 4200 = 2³ × 3¹ × 5² × 7¹

∴ Sum of factors of 4200 = \(\frac{{\left( {{2^{3\; + \;1}} - 1} \right)\left( {{3^{1\; + \;1}} - 1} \right)\left( {{5^{2\; + \;1}} - 1} \right)\left( {{7^{1\; + \;1}} - 1} \right)}}{{\left( {2 - 1} \right)\left( {3 - 1} \right)\left( {5 - 1} \right)\left( {7 - 1} \right)}}\)

⇒ \(\frac{{\left( {16 - 1} \right)\left( {9 - 1} \right)\left( {125 - 1} \right)\left( {49 - 1} \right)}}{{\left( 1 \right)\left( 2 \right)\left( 4 \right)\left( 6 \right)}}\)

⇒ \(\frac{{15\; \times \;8\; \times \;124\; \times \;48}}{{1\; \times \;2\; \times \;4\; \times \;6}}\)

⇒ 15 × 992

⇒ 14880