The hcf and lcm of two numbers are 32 and 192 respectively. If the square of one of the numbers is a perfect cube, find the numbers.

Question

The hcf and lcm of two numbers are 32 and 192 respectively. If the square of one of the numbers is a perfect cube, find the numbers. Please help.

Answer 1

 hcf = 32

lcm = 192

 

the numbers must be a multiple of 32

32, 64, 96, 128, 160, 192

 

The square of one of the numbers

is a perfect cube

32² = 1024

(1024)^1/3 ≈ 10.08

 

64² = 4096

(4096)^1/3 = 16

 

64 is one of the numbers

 

multiples of 32: 32, 64, 96, 128, 160, 192

multiples of 64: 64, 128, 192

multiples of 96: 96, 192

 

the lowest common multiple of

64 and 96 is 192

 

Greatest common factor of

64 and 96 is 32

 

Your numbers are 64 and 96

 

Answer 2

DIVISION METHOD

 

In this method divide the largest number by the smallest number of the given numbers until the remainder is zero. The last divisor will be the HCF of given numbers.

 

Divide the largest by the smallest:
       192/32 = 6 Remainder 0

       The Remainder is already 0, so the HCF is 32 (the last divisor).

PRIME FACTORIZATION METHOD

Given natural numbers are written as the product of prime factors. To obtain highest common factor multiply all the common prime factors with the lowest degree (power).

 

Write as product of prime factors:
        32 = 2*2*2*2*2 = 2⁵
192 = 2*2*2*2*2*2*3 = 2⁶*3¹

Lowest common degree of 2 is 5 and the degree (exponent) for 3 is zero.
             The HCF is 2⁵*3⁰ = 2⁵*1 = 32.