Fundamental

[john123]

If P = (99^x - 99^y), where x & y can be any integer. When P is divided by 100, How many different types of reminders we can find?

a) 3 b) 2 c)1 d) 0 e) none of these

[pemdas]

99 can be expressed as 9*11 and ((9*11)^x-(9*11)^y)/100 is what interests us
9 raised to the different powers returns a unit's digits 1 or 9; 9 or 1 multiplied by 11 raised to any power remain their values
thus, the remainders can be 0 (or no remainder), 98, 1 resuming with 2 remainders

a
[spoiler]
my only concern is we are asked to find the types of remainders and not actual remainders, because 0 is when we have no remainder, but i include this in my final answer. Without 0 we have two types of remainders, with 0 we have 3 types.[/spoiler]

[LalaB]

pemdas, I didnt get it
lets assume x=3 y =2, then -

11(9^3-9^2)/100 =11*9^2*8/100=7128/100

according to ur answer choice, what is the remainder of 7128/100?

[rijul007]

pemdas, I didnt get it
lets assume x=3 y =2, then -

11(9^3-9^2)/100 =11*9^2*8/100=7128/100

according to ur answer choice, what is the remainder of 7128/100?

check it again.. do you see what blunder you've done?

[LalaB]

pemdas, I didnt get it
lets assume x=3 y =2, then -

11(9^3-9^2)/100 =11*9^2*8/100=7128/100

according to ur answer choice, what is the remainder of 7128/100?

check it again.. do you see what blunder you've done?

oh i c

9^2*11^2(9*11-1)/100=98*9^2*11^2/100

[rijul007]

Divide 99 by 100 you get -1 or 99 as a remainder

99^x/100 => -1^x would be the ramainder

99^y/100 => -1^y would be the remainder

(99^x+99^y)/100 => remainder (-1)^x - (-1)^y

when x and y are even
remainder = 1-1 = 0

when x is even and y is odd
remainder = 1-(-1) = 2

when x is odd and y is even
remainder = -1-1 = -2 or 98

when x and y are odd
remainder = -1-(-1) = 0

Number of different types of remainder = 3 {-2,0,2}

Option A

[santhoshsram]

Dumb question. Can someone explain what "type of remainder" means?

[santhoshsram]

Divide 99 by 100 you get -1 or 99 as a remainder

I didn't quite follow this part, remainder -1 when 99 is divided by 100. Can you please explain.