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

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