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Find the remainder when is divided by 81.
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It is clear that the mathematical meaning of this problem is not to follow literally
the written formula.
Its meaning is to decrease / (to reduce) the degrees and values of participating numbers
to make calculations easier using standard properties of operations of modular arithmetic.
Following this idea, I write 40 = 36 + 4,
= .
Next we should apply the Newtonian binomial formula.
It will give the sum of the terms , k = 0, 1, 2, 3, . . . , 13.
All the terms with k >= 2 will be zero by modulo 81, since 36 = 9*4.
Therefore, we can exclude all these terms from our consideration.
So, the terms under our consideration are the terms with k= 0 and k= 1, or
+ = + .
This expression is easy to calculate using a regular calculator or Excel spreadsheet;
its value is 7918845952.
Finally, 7918845952 mod 81 is 22 (use long division or Excel function mod)
So, the ANSWER is 22.
Solved.
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Throwing pebbles into the water from a bridge, look at the circles
they form; otherwise your exercises will be empty deals.