Question

if the h.c.fof two numbers 2^4×3^p×5 and 2^q×3^2×43 is 24,then the sum of two numbers is

Answer 1

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Answer 2

Answer:

The sum of the two numbers is 3336.

Step-by-step explanation:

The two numbers are: 2⁴ * $3^{p}$ * 5  and $2^{q}$ * 3² *43

Given the HCF = 24

2⁴ * $3^{p}$ * 5 = 2 * 2 * 2 * 2 * 3 * 3 * 3 * . . . .(p times) * 5

$2^{q}$ * 3² *43  = 2 * 2 * 2 * . . . . . .(q times) * 3 * 3 * 43

Therefore, the HCF will be $2^{x} * 3^{y}$ = 24

As 24 = 2 * 2 * 2 * 3

=>   $2^{x} * 3^{y}$ = 2³ * 3¹

On equating, x = 3 and y = 1

Therefore, the numbers become,

As 3 is a factor of 24 one time, it means that the highest number of 3 common to both the numbers is 1.

3 is already present two times in  $2^{q}$ * 3² *43

=> 3 will be present one time in 2⁴ * $3^{p}$ * 5

Therefore, p = 1

As 2 is a factor of 24 three time, it means that the highest number of 2 common to both the numbers is 3

2 is already present four times in 2⁴ * $3^{p}$ * 5

=> 2 will be present three times in  $2^{q}$ * 3² *43

Therefore, q = 3

Therefore, the numbers are: 2⁴ * $3^{p}$ * 5 = 2⁴ * 3¹ * 5 = 240

                                      and  $2^{q}$ * 3² *43 =  2³ * 3² *43 = 3096

Therefore, the sum of numbers = 240 + 3096

                                                     = 3336