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1
Sum of all possible permutations of n distinct digits
$$(n-1)!\times\ (sum\ of\ n\ digits)\times\ (11111...n\ times)$$
2
If the number can be expressed as $$N=2^p\times\ a^q\times\ b^r$$ , then the number of even factors of $$N$$
$$p(1+q)(1+r)$$
3
If the number can be expressed as $$N=a^p\times\ b^q\times\ c^r$$ , then the sum of the factors of $$N$$
$$\frac{a^{p+1}-1}{a-1}\times\ \frac{b^{q+1}-1}{b-1}\times\ \frac{c^{r+1}-1}{c-1}$$
4
The number of positive integral solutions of the equation $$x^2-y^2=k$$ ,where k is odd and a perfect square
$$ \frac{\ \text{(total number of factors of k)-1}}{2}$$
5
The number of positive integral solutions of the equation $$x^2-y^2=k$$ , where k is odd and not a perfect square
$$ \frac{\ \text{(total number of factors of k)}}{2}$$
6
Number of digits in $$a^b$$
$$\left[b\log_ma\right]+1$$ {where m= base of the number, and [.] denotes greatest integer function}
7
The product of the factors of $$N$$
$$N^{\frac{a}{2}}\ \text{where a= number of factors}$$
8
$$a^n+b^n$$ is divisible by $$a+b$$, if
$$n\ =\ \text{odd}$$
9
$$a^n-b^n$$ is divisible by $$a+b$$, if
$$n\ =\ \text{even}$$
10
If $$a+b+c =0$$, then
$$a^3+b^3+c^3=3abc$$
11
Highest power of $$n$$ in $$m!$$
$$\left[\frac{m}{n}\right]+\left[\frac{m}{n^2}\right]+\left[\frac{m}{n^3}\right]+....$$
12
If m, n are coprime to each other, then $$\left(m^{Φ\left(n\right)}mod\ n\right)$$ = ?
$$1$$
13
Remainder when $$(p-1)!$$ is divided by $$p$$, where $$p$$ is a prime
$$p-1$$
14
Remainder when $$a^{p-1}$$ is divided by $$p$$, where $$p$$ is a prime
$$1$$
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