Factor Theory Concept ⇒
Consider a composite number n
⇒ n = aᵖbᑫcʳ………..
where a, b, c are prime numbers & p, q, r are natural numbers.Then
Number of factors of n is given by:-
Proof:- A factor of above number has power of a as a⁰ or a¹ or a² or a³ or ……….. aᵖ
similarly, A factor of above number has power of b as b⁰ or b¹ or b² or b³ or ……….. bᑫ
A factor of above number has power of c as c⁰ or c¹ or c² or c³ or ……….. cʳ
Hence factors having a⁰ or any value of a can be (p + 1)
similarly, factors having b⁰ or any value of b can be (q + 1)
factors having c⁰ or nay value of c can be (r + 1)
& when we take any combination of these then total number of factors = (p + 1)(q + 1)(r + 1)……..
where 1 & the number itself is included.
Example:- Find the total number of factors of 2⁴3²5³7⁵.
Solution:-
∴ Total numbers of factors = 5 × 3 × 4 × 6
= 360 Answer
Example:- Find the number of odd factors of 2⁴3²5³7⁵
Solution:-
∴ Total number of odd factors = 1×3×4×6
= 72 Answer
Example:- Find the number of even factors of 2⁴3²5³7⁵
Solution:-
∴ Total number of even factors = 4×3×4×6
= 288 Answer
Example:- Find the number of factors of 2⁴3²5³7⁵ that are divisible by 10.
Solution:-
∴ Total number of factors divisible by 10 = 4×3×3×6
= 216 Answer
Example:- Find the number of factors of 2⁴3²5³7⁵ that are not divisible by 10.
Solution:- required answer = Total factors – factor divisible by 10
=360 – 216
=144 Answer
Example:- Find the number of factors of 2⁴3²5³7⁵ that are divisible by 100.
Solution:-
∴ Total number of ways = 3×3×2×6
= 108 Answer