Modular Again
Is 4 4 9 2 1 2 5 m o d 6 the same as 1 6 5 m o d 6 ?
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2 solutions
You are right. Also, we can see that 4492125=(165)^3 divided by 6 gives the remainder 3.
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My previous comment just equivalent to 4 4 9 2 1 2 5 ≡ 3 m o d 6 & similarly 1 6 5 ≡ 3 m o d 6
Where a ≡ b m o d n
Means, n ∣ a − b
In fact, 4 4 9 2 1 2 5 ≡ 9 m o d 6
4 4 9 2 1 2 5 ≡ 1 5 m o d 6
4 4 9 2 1 2 5 ≡ 2 1 m o d 6
4 4 9 2 1 2 5 ≡ 2 7 m o d 6 ...so on.
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In fact,
For any positive integer n, n^3 mod 6 = n mod 6.
That was my goal in this problem.
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@Hana Wehbi – Yes. Nice observation.
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@Akash Patalwanshi – You are welcome. Nice solutions too, the ones you provided.
Yes, you are right, but then all of them = 3 mod 6. We are going to reach the same result.
4
4
9
2
1
2
5
m
o
d
6
=
1
6
5
m
o
d
6
=
3
.
So, the answer is
True
.
Yes, it is true but I will still provide the solution that works for all n later on.
4 4 9 2 1 2 5 m o d 6 = 3 . Since when 4 4 9 2 1 2 5 divided by 6 gives the r e m a i n d e r 3 .
Similarly one can see 1 6 5 m o d 6 = 3