what will be the remainder when 2^69 is divided by 9
please explain
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Hopefully this image helps u!!
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Kartik0071:
thanksss a lot
but how can you cancel( 9-1)23/9
It's the basic remainder rule whenever you get such great no then try to find out the no nearest to the divisor as I found 2^3 is 8 which can be written as 9-1 so this can possibly cancel the 9 by 9
If you want I can give you an another example for the same!!
yes plsss..
How do I send the image of working?
Ok for say the question is 7^80/50 the remainder is what? Then find the power of 7 such that nearest to the 50's multiple 7^2 makes it 49 now the working:
(7^2)^40/50 =(49)^40/50 now you can write 49 as (50-1)^40/50 cancel the 50 days mathematically correct remaining is (-1)^40 even power makes -1 to 1 so that's the remainder if the power was odd then u have to add the divisor to that -1 the answer arrives is your remainder hope it helps you to clear the concept!! And this the way to get through such remainder suna
okkk thank
Answered by
1
Hey there !!!!!!!!
2⁶⁹ =(2³)²³
Now 2³ =8 =9-1
So,
2⁶⁹=(9-1)²³
(9-1)²³ is a binomial expansion of the form ( x+y)ⁿ
Expansion of (x+y)ⁿ = ⁿC₀xⁿy⁰+ⁿC₁xⁿ⁻¹y¹+ⁿC₂xⁿ⁻²y².........+
Comparing (9-1)²³ with (x+y)ⁿ and expanding it
x=9 y=-1
(9-1)²³ =23C₀9²³(-1)⁰+23C₁(9)²³⁻¹(-1)¹..................+23C₂₃(9)²³⁻²³(-1)²³
In the above expansion except the last term all the terms are divisible by 9.As all these terms are a factor of 9 as they contain powers of 9.
Last term which is 23C₂₃(9)²³⁻²³(-1)²³=23C₂₃(9)⁰(-1)²³ =23C₂₃*1*(-1)²³ is not divisible by 9 as it does not contain powers of 9.
23C₂₃ = 1
So 23C₂₃(9)²³⁻²³(-1)²³ = -1
So when we divide (9-1)²³ with 9 "-1" is the remainder.
But remainder can't be negative.
So adding and subtracting 8 to -1
R=-1-8+8 =-9 +8
First part of R i.e, -9 is divisible by 9 so remainder is 8.
Hope this helped you........
2⁶⁹ =(2³)²³
Now 2³ =8 =9-1
So,
2⁶⁹=(9-1)²³
(9-1)²³ is a binomial expansion of the form ( x+y)ⁿ
Expansion of (x+y)ⁿ = ⁿC₀xⁿy⁰+ⁿC₁xⁿ⁻¹y¹+ⁿC₂xⁿ⁻²y².........+
Comparing (9-1)²³ with (x+y)ⁿ and expanding it
x=9 y=-1
(9-1)²³ =23C₀9²³(-1)⁰+23C₁(9)²³⁻¹(-1)¹..................+23C₂₃(9)²³⁻²³(-1)²³
In the above expansion except the last term all the terms are divisible by 9.As all these terms are a factor of 9 as they contain powers of 9.
Last term which is 23C₂₃(9)²³⁻²³(-1)²³=23C₂₃(9)⁰(-1)²³ =23C₂₃*1*(-1)²³ is not divisible by 9 as it does not contain powers of 9.
23C₂₃ = 1
So 23C₂₃(9)²³⁻²³(-1)²³ = -1
So when we divide (9-1)²³ with 9 "-1" is the remainder.
But remainder can't be negative.
So adding and subtracting 8 to -1
R=-1-8+8 =-9 +8
First part of R i.e, -9 is divisible by 9 so remainder is 8.
Hope this helped you........
Bhai! Gazzab! Awesome answer! ^^
thanks bhai :)
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