The number is less than $100$, but five times the number is greater than $100$, so it must be between $20$ and $100$.

Reversing the digits makes a prime number, so the first digit must be $1$, $3$, $7$ or $9$, as all two-digit prime numbers are odd and not divisible by $5$. The number must be at least 20, so this rules out $1$.

Since the digits add to a prime number, the possibilities are:

First Digit | Second Digit |
---|---|

$3$ | $2,4,8$ |

$7$ | $4,6$ |

$9$ | $2,4,8$ |

The number must be one more than a multiple of $3$, so the digits must sum to give one more than a multiple of $3$, as multiples of $3$ have digit sums that are multiples of $3$. This leaves $34$, $76$ and $94$.

The number must have exactly one prime digit, which rules out $94$.

The number must have exactly four factors. $34$ has $1$, $2$, $17$ and $34$, but $76$ has $1$, $2$, $4$, $19$, $38$ and $76$.

Therefore the number is $34$.

*This problem is taken from the UKMT Mathematical Challenges.*