What will be the reainder when (1000000)^3 is divided by 143?
Answers
Answer:remainder of
(1 000 000)^3 / 143
=10^18 mod 143
1001=143(7)
10^3=1000≡-1 mod 143
10^18=(-1)^6≡1 mod 143
remainder is 1.
Note:
143=11(13)
10≡-1 mod 11
10≡-3 mod 13
10^3≡-1 mod 11
10^3≡-27 mod 13 ≡-1 mod 13
Therefore 10^3 ≡ -1 mod 13
Request to not use mod.
1001 = 143(7)
1000 = 143(7)-1
10^3 = 7(143) -1
10^18=(7(143)-1)^6
Use the binomial theorem.
The first 6 terms are multiples of 143. The last term is (-1)^6=1.
10^18=Q(143) + 1
Q is some integer.
Therefore the remainder on dividing by 143 is 1
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Given:
(1000000)^3 is divided by 143.
To Find:
The reainder when (1000000)^3 is divided by 143.
Solution:
(1000000)^3 / 143.
= 10^18 mod 143.
1001 = 143(7).
10^3 = 1000 ≡ -1 mod 143.
10^18 = (-1)^6 ≡ 1 mod 143.
The remainder is 1.
Here,
143 = 11(13).
10 ≡ -1 mod 11.
10 ≡ -3 mod 13.
10^3 ≡ -1 mod 11.
10^3 ≡ -27 mod.
13 ≡ -1 mod 13.
Therefore,
10^3 ≡ -1 mod 13.
Now,
1001 = 143(7).
1000 = 143(7) - 1.
10^3 = 7(143) - 1.
10^18 = (7(143) - 1)^6.
Using the binomial theorem,
The first 6 terms are multiples of 143. The last term is (-1)^6 = 1.
10^18 = Q(143) + 1
Q is any integer.
Hence, the remainder on dividing by 143 is 1.