- #1

Math100

- 760

- 206

- Homework Statement
- Determine all integers ## n ## for which ## \phi(n) ## is a divisor of ## n ##.

- Relevant Equations
- None.

Observe that ## \phi(1)=\phi(2)=1 ##.

This implies ## \phi(1)\mid 1 ## and ## \phi(2)\mid 2 ##.

Thus ## n=1 ##.

Let ## n=p_{r}^{k_{1}}\dotsb p_{s}^{k_{s}} ## be the prime factorization of ## n ##.

Then ## \phi(n)=n\prod_{p\mid n} (1-\frac{1}{p}) ##.

Suppose ## \phi(n)\mid n ##.

Then ## \frac{n}{\phi(n)}=(\prod_{p\mid n} \frac{p}{p-1}) ## for some ## n\in\mathbb{Z^{+}} ##.

Note that the prime divisors of ## n ## must be ## 2 ## or ## 3 ## for ## n>1 ##, because ## p-1 ## is even for ## p\geq 3 ##.

Therefore, all integers ## n ## for which ## \phi(n) ## is a divisor of ## n ## are ## 1, 2^{r} ## and ## 2^{r}3^{j} ## for some ## r, j\in\mathbb{N} ##.

This implies ## \phi(1)\mid 1 ## and ## \phi(2)\mid 2 ##.

Thus ## n=1 ##.

Let ## n=p_{r}^{k_{1}}\dotsb p_{s}^{k_{s}} ## be the prime factorization of ## n ##.

Then ## \phi(n)=n\prod_{p\mid n} (1-\frac{1}{p}) ##.

Suppose ## \phi(n)\mid n ##.

Then ## \frac{n}{\phi(n)}=(\prod_{p\mid n} \frac{p}{p-1}) ## for some ## n\in\mathbb{Z^{+}} ##.

Note that the prime divisors of ## n ## must be ## 2 ## or ## 3 ## for ## n>1 ##, because ## p-1 ## is even for ## p\geq 3 ##.

Therefore, all integers ## n ## for which ## \phi(n) ## is a divisor of ## n ## are ## 1, 2^{r} ## and ## 2^{r}3^{j} ## for some ## r, j\in\mathbb{N} ##.