There are two positive integers M and N.When M is divided by 289, the remainder is 192.When N is divided by 171 the quotient is same but the remainder is 169. Find the remainder when the sum of M and N is divided by 115.
Answers
Step-by-step explanation:
Given:-
There are two positive integers M and N.When M is divided by 289, the remainder is 192.When N is divided by 171 the quotient is same but the remainder is 169.
To find:-
Find the remainder when the sum of M and N is divided by 115.
Solution:-
Given that
M is divided by 289, the remainder is 192
We know that
Dividend = Divisor× Quotient + Remainder
=> M = 289× Q + 192
=> M = 289 Q + 192 ----------------(1)
and
N is divided by 171 the quotient is same but the remainder is 169.
=> N = 171 × Q + 169
=> N = 171 Q + 169 ----------------(2)
On adding (1) & (2) equations then
=> M + N
=> 289 Q + 192 + 171 Q + 169
=> (289 Q + 171 Q ) + ( 192 + 169 )
=> 460 Q + 361
We have , M + N = 460 Q + 361 --------(3)
On dividing (3) by 115 both sides then
=> (M + N ) / 115 = (460 Q + 361 ) / 115
=> (M + N ) / 115 = (460 Q ) /115 + ( 361 / 115 )---(5)
The remainder = 361/ 115
It can be written as
361 = 115 × 3 + 16
(5) can be written as
=> ( M + N ) / 115 = (460 Q)/115 + (345/115) + 16
=> ( M + N ) / 115 = (460 Q+345)/115 + 16
=> ( M + N) / 115 = (4Q +3) + 16
Quotient = 4Q+3
Remainder = 16
Answer:-
The required remainder when the sum of M and N is divided by 115 is 16
Used formula:-
Division Rule :-
- Dividend = Divisor× Quotient + Remainder