# When a two digit number is added to its reverse, the result is 143. The number is 3 less than the sum of the squares of its digits. Calculate this number.

2
by kinzo

number is 94.

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by kinzo

number is 94.

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Let the number be N = a b where b is the units digit. a = the tens digit.

the value of the number* N = 10 * a + b*

number after reversing : b a

the value of the number* after reversing : 10 * b + a *

sum of the two numbers = 10 a + b + 10 b + a = 11 (a + b)

given, 11 ( a + b) = 143 =>* a + b = 13 *--- (1)

sum of squares of digits =* a² + b² = (a+b)² - 2 a b*

given,* N = 10 a + b = (a² + b²) - 3*

=> 10 a + b = (13² - 2 a b) - 3 = 166 - 2 a b

Hence,* 10 a + b + 2 a b = 166 * --- (2)

do (2) - 10 * (1) : - 9 b + 2 a b = 36

so* b (2a -9) = 36 --- (4)* .

substitute value of b from (1) in (4) :

(13 - a) ( 2a - 9) = 36

=> -2 a² + 35 a - 117 = 36

=> 2 a² - 35 a + 153 = 0

discr: 35² - 8 * 153 = 1

a = (35 +- 1)/ 4 = 9 or 8.5

we take the integer value only as a and b are integers less than 10.

* so a = 9 and b = 4 *

so the number is 94.

the value of the number

number after reversing : b a

the value of the number

sum of the two numbers = 10 a + b + 10 b + a = 11 (a + b)

given, 11 ( a + b) = 143 =>

sum of squares of digits =

given,

=> 10 a + b = (13² - 2 a b) - 3 = 166 - 2 a b

Hence,

do (2) - 10 * (1) : - 9 b + 2 a b = 36

so

substitute value of b from (1) in (4) :

(13 - a) ( 2a - 9) = 36

=> -2 a² + 35 a - 117 = 36

=> 2 a² - 35 a + 153 = 0

discr: 35² - 8 * 153 = 1

a = (35 +- 1)/ 4 = 9 or 8.5

we take the integer value only as a and b are integers less than 10.

so the number is 94.

the no. will be 10x+y

according to the question,,10x+y + 10y+x=143

11x+11y=143

by solving we get x+y=13

y=x-13 ..............eq 1

according to the question,,10x+y+3=x²+y² ............................eq 2

substituting eq 1 in eq 2

10x+13-x+3=x²+(13-x)²

by solving we get 2x²-35x+153=0

Solving the quadratic equation

a=2 b=-35 c=153

determinant=b²-4ac

=1225-1224

=1

√D =1

x=-b+√D÷2a or x=-b-√D÷2a

x=9 or x=34/4(ruled out since digits cannot

be fractions)

hence x=9

y=13-9

=4

the number is 94.

hope this helps u...