please answer my question tennetiraj sir.
Answers
Answer:
154
Step-by-step explanation:
Let x = 10a + b and y = 10b + a
x² - y² = m²
10b + a .. (10a + b)²(10b + a)² = m²
:. (10a+b+10b+ a)(10a + b - 10b a) = m²
11(a+b)9(a - b) = m²
:.99 (a + b)(a - b) = m
²
Since a and bare digits from 1 to 9, a + b = 11 and a - b = 1 is the only
a = 6 and b = 5 possibility for m to be an integer.
⇒ x = 65, y = 56, m = 33
:.x+y+m= 65+56 +33 = 154
Step-by-step explanation:
Given :-
Let x and y be the two digits numbers such that y is obtained by the reversing the digits of x and they also satisfy x²-y² = m² for some positive integers m.
To find :-
Find the value of x+y+m ?
Solution :-
Given that
x and y are two digits numbers
Let x = 10a+b
On squaring both sides then
=> x² = (10a+b)²
=> x² = 100a²+20ab+b² -----------(1)
given that
y is obtained by reversing the digits of x
Therefore, y = 10b+a
On squaring both sides then
=> y² = (10b+a)²
=> y² = 100b²+20ab+a² ---------(2)
On Subtracting (2) from (1) then
=>x²-y²
=(100a²+20ab+b²)-(100b²+20ab+a²)
=>x²-y² = 100a²+20ab+b²-100b²-20ab-a²
=>x²-y² = 100a²-a²-100b²+b²
=> x²-y² = (100-1)a²+(-100+1)b²
=> x²-y² = 99a²-99b²
=> x²-y² = 99(a²-b²) --------------(3)
According to the given problem
Given condition is x²-y² = m² -------(4)
From (3)&(4)
=> 99(a²-b²) = m² ---------(5)
The product of 99 and a²-b² is equal to the square number m²
=> 9×11(a²-b²) = m²
=> 3² × 11 (a²-b²) = m²
To get perfect square a²-b² should be 11
=> 3²×11(11)= m²
=> 3²×11² = m² ,a perfect square number
So , a²-b² = 11
=> (a+b)(a-b) = 11
=> (a+b)(a-b) = 11×1
=>(a+b)(a-b) = (6+5)(6-5)
On comparing both sides then
a = 6
b = 5
Now x = 10a+b = 10(6)+5 = 65
y = 10b+a = 10(5)+6 = 50+6 = 56
The two digits numbers are 65 and 56
From (4)
x²-y² = m²
=> (65)²-(56)² = m²
=> 4225-3136 = m²
=> 1089 = m²
=> m² = 1089
=> m² = 33²
=> m = 33
The value of m = 33
The value of x+y+m
On Substituting the values of x,y and m then
=>x+y+m = 65+56+33
=> x+y+m = 154
Therefore, x+y+m = 154
Answer:-
The value of x+y+m for the given problem is 154
Used formulae:-
- (a+b)(a-b) = a²-b²
- The numbers are written as the product of two equal numbers is called a perfect square number.