Key facts on Remainder theorem that you should know

KEY FACTS on Remainder Theorem

These facts are helpful in dealing with large power.

1. If we can express the expression in the form $$\frac{{{{\left( {{\text{ax}} + 1} \right)}^{\text{n}}}}}{{\text{a}}},$$   the remainder will become 1 directly. In such case no matter how large the value of the power n is, the remainder is 1. E.g.
$$\frac{{{{37}^{1234692}}}}{9}$$   ==R==> $$\frac{{{{\left[ {\left( {9 \times 4} \right) + 1} \right]}^{1234692}}}}{9}$$    ==R==> 1^1234692 ==R==>1.

2. $$\frac{{{{\left( {{\text{ax}} - 1} \right)}^{\text{n}}}}}{{\text{a}}}.$$  In such case,
when, n = odd, remainder = -1 = (a-1)
When, n = even, remainder = 1
e.g.
$$\frac{{{{31}^{1234692}}}}{8}$$  ==R==> $$\frac{{\left( {8 \times 4} \right) - 1}}{8}$$   ==R==> -1 ==R==> (8-1) = 7. Where a = 8 and x = 4.

3. If we can express the expression in the form $$\frac{{{{\left( {{\text{ax + m}}} \right)}^{\text{n}}}}}{{\text{a}}},$$  then required remainder is same as $$\frac{{{{\text{m}}^{\text{n}}}}}{{\text{a}}}$$. e.g.
Remainder of $$\frac{{{{67}^{99}}}}{7}$$ ==R==> $$\frac{{{{\left( {63 + 4} \right)}^{99}}}}{7}$$  ==R==> $$\frac{{{4^{99}}}}{7}$$ ==R==> $$\frac{{{4^{96 + 3}}}}{7}$$ ==R==> $$\frac{{{4^3}}}{7}$$ ==R==> 1.

4. $$\frac{{{{\text{a}}^{\text{n}}}}}{{{\text{a}} + 1}}$$ leaves a remainder of
a if n is odd.
1 if n is even.

5. $$\frac{{{{\left( {{\text{a}} + 1} \right)}^{\text{n}}}}}{{\text{a}}}$$  will always gives a remainder 1.

6. aⁿ + bⁿ is always divisible by (a+b) when n is odd. Eg.
(15²³ + 23²³) always divisible by 38 and as well as its multiples.
So,
$$\frac{{{{15}^{23}} + {{23}^{23}}}}{{19}}$$   ==R==> $$\frac{{15 + 23}}{{19}}$$  ==R==> $$\frac{{38}}{{19}}$$ ==R==> 0

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