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Question
A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.
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Solution
Let the require digit be = (10x + y)
Then according to question
(10x + y) = 4(x + y)
(10x + y) = 4x + 4y
10x + y - 4x - 4y = 0
6x - 3y = 0
2x - y = 0
2x = y ................(1)
And, (10x + y) = 2xy .........(2)
Now putting the value of y in equation (2) from (1)
(10x + 2x) = (2x)(2x)
4x2 - 12x = 0
4x(x - 3) = 0
x(x - 3) = 0
So, either
x = 0
Or
x - 3 = 0
x = 3
So, the digit can never be negative.
When x = 3 then
y = 2x = 2 × 3 = 6
Therefore, number
=10x + y
= 10(3) + 6
= 30 + 6
= 36
Thus, the required number be 36.
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