The square root of a number consisting of two digits isequal

to the sum of the digits and is less than the number obtainedby

interchanging the digits by 9. Find the number.

81

Step-by-step explanation:

Let the digits that make up the number be a and b.

Given that the square root of the number is equal to the sum of the digits.

Then,

√ (10a + b) = a + b

Also given that the square root of the number is less than the number obtained by interchanging the digits by 9, then

√ (10a + b) + 9 = 10b + a

Since √ (10a + b) = a + b, then

a + b + 9 = 10b + a

a - a + 9 = 10b - b

9b = 9

b = 1

since √ (10a + b) = a + b

√ (10a + 1) = a + 1

10a + 1 = (a + 1) ²

10a + 1 = a² + 2a + 1

a² + 2a - 10a + 1 - 1 = 0

a² - 8a = 0

a (a - 8) = 0

a = 0 or a = 8

Using a = 8 and b = 1,

the number 10a + b = 10 (8) + 1 = 81.