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A two-digit number is such that the product of its digits is 35. If 18 is added to the number, the digits interchange their places. Find the number.

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This is the Important question based on Linear Equations in two Variables Chapter of R.S Aggarwal book for ICSE & CBSE Board.
Question Number 16 Exercise 3E of RS Aggarwal Solution
Here A two-digit number is such that the product of its digit is given. if a number is added to the number, the digits interchange their places.
You have to find the number.

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  1. Let the unit digit be ‘x’
    Let the ten’s digit be ‘y’.
    Original number will be
    10x ten’s digit+ one’s digit
    =10x+y
    On interchanging of digits,
    Let unit’s digit be ‘y’.
    Let ten’s digit be ‘x’.
    New number will be
    10 x ten’s digit + one’s digit
    =10y+x
    According to question,
    Product of digit of two digit number = 35
    So, it becomes,
    xy=35———–1
    If we add 18 to the number the new number obtained is number formed by interchange of digits.
    10x+y+18=10y+x
    10x-x+18=10y-y
    9x+18=9y
    18=9(y-x)
    18/9=y-x
    2=y-x
    y=2+x—————2
    Put the value of eq(2) in Eq(1).
    xy=21
    x(2+x)=35
    x^2+2x=35
    x^2+2x-35=0
    x^2+7x-5x-35=0
    x(x+7)-5(x+7)=0
    (x+7)(x-5)=0
    x+7=0
    x-5=0
    x=-7 ,x=5
    x=-7 will be rejected.
    So, x=5 and
    xy=35
    5y=35
    y=35/5
    y=7
    Hence, original number will be
    10x+y=10×5+7=50+7=57
    New number will be
    10y+x=10×7+5=70+5=75
    Therefore, Numbers are 57 and 75.

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