Question

The sum of two numbers is 204 and their HCF is 17. Find all the possible pairs of such numbers. Answer will be 119 but please explain with process ​

Answer 1

Answer:

your answer

Step-by-step explanation:

Let the two numbers be x and y respectively.

It is given that the sum of the two numbers is 204, therefore,

     

x+y=204

Also 17 is their HCF, thus both numbers must be divisible by 17.

So, let x=17a and y=17b, then  

17a+17b=204

⇒17(a+b)=204

⇒a+b=  

17

204

​  

 

⇒a+b=12

           

Therefore, required possible pair of values of x and y which are prime to each other are (1,11) and (5,7).

Thus, the required numbers are (17,187) and (85,119).

Hence, the number of possible pairs is 2.

Answer 2

Answer:

$\huge\underline\bold {Answer:}$

Since the HCF is 17, the number must be of the form 17a and 17b, where a and b are prime to each other.

Given, 17a + 17b = 204

=> a + b = 12

The possible pairs of numbers whose sum is 12 are 1, 11; 2, 10; 3, 9; 4, 8; 5, 7; 6, 6

The only pairs of numbers which are prime to each other are 1, 11; 12, 10; 3, 9; 4, 8; 5, 7; 6, 6

Therefore, the required pairs are 17 × 1 = 17

and 17 × 11 = 187

and 17 × 5 = 85

and 17 × 7 = 119.