The difference between two positive integers is 2 and the difference between their cubes is 56. Find the numbers.
Let the numbers be x and y.
According to question
x−y=2 …………..(1)
x3−y3=56 …………..(2)
From (1)
x=y+2
Substitute the value of x in equation (2)
(y+2)3−y3=56
⇒y3+8+6y2+12y−y3=56 .................. Using (a+b)3=a3+3a2b+3ab2+b3
⇒6y2+12y−56+8=0
⇒6y2+12y−48=0
⇒6(y2+2y−8)=0
⇒y2+4y−2y−8=0
⇒y(y+4)−2(y+4)=0
⇒(y+4)(y−2)=0
y+4=0 y−2=0
y=−4 y=2
Y cannot be negative
∴y=2
Then x=y+2
⇒x=2+2
⇒x=4.