If we multiply a certain two digit number by the sum of its digit we get 405 if we multiply - Maths - Mathematical Reasoning

Question

If we multiply a certain two digit number by the sum of its digit
we get 405 if we multiply the number consisting of the same digits
written in the reverse order by the sum of the digits wicked 486
find the number

Ashok Kumar Vardhan · Accepted Answer

Dear student,
Let the digit in units' place be y and the digit in tens' place be x
Hence the number is 10x+y⇒Reverse of number=10y+xGiven,The number multiplied by sum of its digits=405⇒10x+y(x+y)=405⇒10x²
+ 10xy+xy+y² = 405⇒10x² + 11xy+y² =
405⇒y² = 405 - 10x² -
11xy→1Also,Reverse of number multiplied by sum of its digits=486⇒(10y+x)(x+y)=486⇒10xy+
10y² +x² +xy= 486⇒11xy+x² + 10y² =
486→2Substituting 1 in 2,⇒11xy+x² +
10(405 - 10x² - 11xy) = 486⇒11xy+x² + 4050 -
100x² - 110xy= 486⇒-99x² - 99xy+ 4050 =
486⇒99x² + 99xy= 4050 - 486⇒99(x² +xy) =
3564⇒x² +xy=356499⇒x² +xy= 36⇒x(x+y) =
36⇒x+y=36x   →3Substituting 3 in 1,⇒(10x+y)36x =
405⇒360x+36y= 405x⇒36y= 405x- 360x⇒36y=
45x⇒xy=3645⇒xy=45⇒x:y::4:5Hence,the 2 digit number is 10x+y⇒10(4)+5⇒40+5⇒45

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If we multiply a certain two digit number by the sum of its digit we get 405 if we multiply the number consisting of the same digits written in the reverse order by the sum of the digits wicked 486 find the number