# Numbers with Euler Phi Value of 72

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## Example of Use of Euler $\phi$ Function

There are $17$ positive integers for which the value of the Euler $\phi$ function is $72$:

- $73, 91, 95, 111, 117, 135, 146, 148, 152, 182, 190, 216, 222, 228, 234, 252, 270$

## Proof

\(\ds 72\) | \(=\) | \(\ds \map \phi {73}\) | Euler Phi Function of Prime | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {91}\) | $\phi$ of $91$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {95}\) | $\phi$ of $95$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {111}\) | $\phi$ of $111$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {117}\) | $\phi$ of $117$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {135}\) | $\phi$ of $135$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {146}\) | $\phi$ of $146$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {148}\) | $\phi$ of $148$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {152}\) | $\phi$ of $152$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {182}\) | $\phi$ of $182$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {190}\) | $\phi$ of $190$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {216}\) | $\phi$ of $216$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {222}\) | $\phi$ of $222$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {228}\) | $\phi$ of $228$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {234}\) | $\phi$ of $234$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {252}\) | $\phi$ of $252$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \map \phi {270}\) | $\phi$ of $270$ |

$\blacksquare$

## Sources

- 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $72$