For natural number N = 24x33x52 we have (A) The numer of factors of 'N' is 72 (B) The number of odd factors of 'N' is 12 (C) The number of even factors of Nis 60 (D) Product of factors of N is 578
Answers
Option D is the false statement
Step-by-step explanation:
Given :-
Natural number N = 24x33x52
To find :-
Verify the following :
(A) The numer of factors of 'N' is 72
(B) The number of odd factors of 'N' is 12
(C) The number of even factors of Nis 60
(D) Product of factors of N is 578
Solution :-
Given number N = 24×33×52
24 can be written as 2×2×2×3
24 = 2³×3¹
33 can be written as 3×11
33 = 3¹×11¹
52 can be written as 2×2×13
52 = 2²×13¹
Now
N = 2³×3¹×3¹×11¹×2²×13¹
=> N = (2³×2²)×(3¹×3¹)×11¹×131
=> N = 2⁵×3²×11¹×13¹
It is in the form of a^p × b^q×c^r×d^s
We know that
The total number of factors = (p+1)(q+1)(r+1)(s+1)
Total number of all factors of N
= (5+1)(2+1)(1+1)(1+1)
=> 6×3×2×2
=> 72
and
Odd factors of N = (2+1)(1+1)(1+1)
Since ,3,11,13 are odd
=> 3×2×2
=> 12
Number of even factors = Total number of factors -number of odd factors
=> 72-12
=> 60
and
Product of all factors
=N^ (Total number of factors/2)
= (24×33×52)^(72/2)
=> (24×33×52)^36
=> 41184^36
Answer :-
D is not true statement
Used formulae:-
A number N = a^p× b^q×c^r×d^s ,where a,b,c,d are primes then
→ Total number of factors of N = (p+1)(q+1)(r+1)(s+1)
→ Total number of even factors = Total number of all factors - Total number of odd factors
→ Product of all factors of N
= N^ (Total number of factors/2)
Step-by-step explanation:
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