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प्रश्न
A two-digit number is 4 times the sum of its digits and twice the product of the digits. Find the number.
उत्तर
Let the digits at units and tens place of the given number be x and y respectively. Thus, the number is `10 y + x`.
The number is 4 times the sum of the two digits. Thus, we have
` 10 y +x =4( x + y)`
` ⇒ 10y + x = 4x + 4y`
`⇒ 4x + 4y -10y -x =0 `
` ⇒ 3x -6y =0`
`⇒ 3(x - 2y)=0`
` ⇒ x- 2y =0`
` ⇒ x = 2y`
After interchanging the digits, the number becomes `10x + y`.
The number is twice the product of the digits. Thus, we have `10y+x=2xy`
So, we have the systems of equations
` x = 2y,`
` 10y +x =2xy`
Here x and y are unknowns. We have to solve the above systems of equations for xand y.
Substituting `x = 2y` in the second equation, we get
` 10y + 2y = 2xx2yxxy`
` ⇒ 12y = 4y^2`
` ⇒ 4y^2-12y =0`
` ⇒ y ( y -3)=0`
` ⇒ y =0` OR `y = 3`
Substituting the value of y in the first equation, we have
Hence, the number is `10 xx 3+6= 36.`
Note that the first pair of solution does not give a two digit number.
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