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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
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B
Odd positive integers
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C
Prime but not odd
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D
Odd but not prime
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Solution

The correct option is C Odd positive integers
Let x and y = the numbers
 

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

(xy)(x2y2)=352(x−y)(x2−y2)=352   ←   Equation (2)
 

(x+y)(x2+y2)(xy)(x2y2)=5500352(x+y)(x2+y2)(x−y)(x2−y2)=5500352

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

x2+y2(xy)2=1258x2+y2(x−y)2=1258

8x2+8y2=125(x22xy+y2)8x2+8y2=125(x2−2xy+y2)

117x2150xy+117y2=0117x2−150xy+117y2=0

(13x9y)(9x13y)=0(13x−9y)(9x−13y)=0
 

For 13x - 9y = 0
y=139xy=139x   ←   Equation (3)
 

From Equation (2)
(x139x)[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



Mathematics

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