Question

If 764xy is divisible by 90 then what is the value of x+y

Answer 1

$\underline{\underline{\Huge{\textbf{Question:}}}}$

If 764xy is divisible by 90 then what is the value of x+y



$\underline{\LARGE{\textsf{Concept\: Behind:}}}$

$\blacksquare$ If a number 'P' is divisible by another number 'Q' , then $\textsf{any number divisible by 'P' will also be divisible by 'Q' and all factors of 'Q'.}$

$\blacksquare\: \underline{\textbf{Divisibility\: Rule\: of\: 9: }}$
The $\textit{divisibility rule of 9}$ states that a number is divisible by 9 if and only if $\textsf{sum of it's digits is divisible by 9}$.

$\blacksquare\: \underline{\textbf{Divisibility\: Rule\: of\: 10: }}$
The $\textit{divisibility rule of 10}$ states that a number is divisible by 10 if and only if the number has $\textsf{zero as it's unit's digit}$.



$\underline{\underline{\Huge{\textbf{Solution:}}}}$

• If the number 764xy is divisible by 90, then the number 764xy is also divisible by all factors of 90.

Hence, 764xy is also divisible by 10 and 9.

If the No. 764xy is divisible by 10, then Unit's digit y = 0

$\therefore y = 0$

• If the Number 764xy is divisible by 9, then sum of it's digits is also divisible by 9

Sum of it's digits = 7+6+4+x+0 = 17+x

The Nearest multiple of 9 is 18.

If the sum of it's digits is to be divisible by 9, then the value of x is 1.

$\therefore x=1$

$\therefore x+y = 1+0 = 1$



$\underline{\huge{\textbf{Method-2:}}}$

$\begin{array}{r|l} &849\\ \cline{1-2} & \\ 90 & 7\: 6 \: 4\: x\: y \\ & 7\: 2\: 0 \\ \cline{2-2} & \: \: \: \: 4\: 4\: x\\&\: \: \: \: 3\: 6\: 0\\\cline{2-2} &\:\: \: \: \: \: \: 8\: x\: y\\&\:\: \: \: \: \: \: 8\:1\: 0\\\cline{2-2}&\quad \quad \: 0 \\ \end{array}$

In the third part of the division, the three digit number that is divisible by 90 and starting with 8 is 810.

Since, the number is divisible by 90,

$\implies 8xy-810= 0$

$\implies 8xy = 810$

$\therefore x = 1\: and\: y= 0$

$\therefore x+y = 1+0 = 1$



$\therefore$

$\blacksquare \underline{\Large{\textbf{The\: Value\: of\: x+y = 1}}}$

Answer 2

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