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What is $x$ if $x-\frac{13951396^{2016}}{90} $ is an integer: $\frac{46}{90}$ or $\frac{47}{90}$or $\frac{35}{90}$

Is there any simple way to solve this?

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    You need to calculate $13951396^{2016}\pmod{90}$. – 2017-01-27
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    why? please explain more – 2017-01-27
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    You have to do that in order to see how far that fraction is from being an integer. – 2017-01-27
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    Hint: even numbers stay even mod $90,\,$ so it can only be $46$, not $47$ or $35$ – 2017-01-27
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    @Hamid If you don't understand the prior comment (or my answer) using parity (even/odd) arithmetic then please feel welcome to ask for elaboration. None of the elaborate calculations in your accepted answer are required. – 2017-01-27
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    @BillDubuque Thank you , I nearly understood your commnet – 2017-01-27

3 Answers 3

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We have, modulo 90:

$13951396\equiv(1+3+9+5+1+3+9)\times 10+6=316\equiv 46$

To get this we cast out nines from the tens and larger digits and then tack on the units digit.

Then

$46^2=2116\equiv 46$ mod $90$

so residue $46$ is idempotent, thus we are sure that the numerator in the original fraction has that residue.

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$\begin{align}{\rm Note}\ \ x \, =\ &\dfrac{a}{90} =\, n + \dfrac{(2j)^{\large k}}{90},\ \ k>0\\[0.3em] \Rightarrow\ a &\,=\, 90 n + (2j)^{\large k}\,\ \text{is even, so $\,a\,$ can only be $46$, not $47$ or $35$}\end{align}$

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$$\dfrac{13951396^{2016}}{90}=\\ \dfrac{13951396^{2015}.13951396}{90}=\\ \dfrac{13951396^{2015}.6975698}{45}=\\$$ so you can check by mod 45 $$13951396\equiv1 \space (mod \space 45)\\ 13951396^{2015} \equiv 1 \space (mod \space 45)\\6975698 \equiv23 \space (mod \space 45)\\ \to 13951396^{2015}.6975698\equiv1\times 23 \space (mod \space 45)\\ 13951396^{2015}.6975698=45q+23$$ so $$\dfrac{13951396^{2015}.6975698}{45}=\dfrac{45q+23}{45}=\dfrac{90q+46}{90}$$ now $$x-\dfrac{90q+46}{90} $$ is integer $$x=\dfrac{46}{90}\\\dfrac{46}{90}-\dfrac{90q+46}{90} \in Z$$

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    This is much wasted effort since a simple *parity* check solves the problem with trivial mental aritmetic. – 2017-01-27
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    @Bill Dubuque : you are right ,but I solve it not just as a multiple choice test ... – 2017-01-27
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    Presumably it was presented as choices precisely to get the student to recognize that it can be solved quickly and simply by parity arithmetic. If this were a test then proceeding as above would waste too much time. – 2017-01-27
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    Answers are not just for the asker. I was happy to see how this could be done. – 2017-01-27