Question:
A two digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits interchange their places. Find the number.
Answer:
Let the two digit number is 10x + y
Given, the product of its digits = 18
=> xy = 18 ............1
Again, when 63 is subtracted from the number, the digits interchange their places
=> 10x + y - 63 = 10y + x
=> 10x + y - 10y - x = 63
=> 9x - 9y = 63
=> 9(x - y) = 63
=> x - y = 63/9
=> x - y = 7 .............2
Now, (x + y)2 = (x - y)2 + 4xy
=> (x + y)2 = 72 + 4 * 18
=> (x + y)2 = 49 + 72
=> (x + y)2 = 121
=> x + y = √121
=> x + y = ±11
Case 1. when x - y = 7 and x + y = 11
After solving it, we get
x = 9, y = 2
Case 2. when x - y = 7 and x + y = -11
After solving it, we get
x = -2, y = -9, wehich is not possible.
So, x = 2, y = 7
So, the number = 10*9 + 2
= 90 + 2
= 92
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