Question

Two numbers are such that their sum multiplied by the sum of their squares is 5500 and their difference multiplied by the difference of the squares is 352. Then the numbers are

(A) Prime numbers only

(B) Odd positive integers

(C) Prime but not odd

(D) Odd but not prime

Answer 1

Answer:

(B) Odd positive integers

Step-by-step explanation:

Let x and y = the numbers

 

(x+y)(x2+y2)=5500(x+y)(x2+y2)=5500   ←   Equation (1)

(x−y)(x2−y2)=352(x−y)(x2−y2)=352   ←   Equation (2)

 

(x+y)(x2+y2)(x−y)(x2−y2)=5500352(x+y)(x2+y2)(x−y)(x2−y2)=5500352

(x+y)(x2+y2)(x−y)(x−y)(x+y)=1258(x+y)(x2+y2)(x−y)(x−y)(x+y)=1258

x2+y2(x−y)2=1258x2+y2(x−y)2=1258

8x2+8y2=125(x2−2xy+y2)8x2+8y2=125(x2−2xy+y2)

117x2−150xy+117y2=0117x2−150xy+117y2=0

(13x−9y)(9x−13y)=0(13x−9y)(9x−13y)=0

 

For 13x - 9y = 0

y=139xy=139x   ←   Equation (3)

 

From Equation (2)

(x−139x)[x2−(139x)2]=352(x−139x)[x2−(139x)2]=352

(−49x)(−8881x2)=352(−49x)(−8881x2)=352

(−49x)(−8881x2)=352(−49x)(−8881x2)=352

352729x3=352352729x3=352

x3=729x3=729

x=9x=9       answer

 

From Equation (3)

y=139(9)y=139(9)

y=13y=13