# Euler problems/51 to 60

## Contents

## Problem 51

Find the smallest prime which, by changing the same part of the number, can form eight different primes.

Solution:

```
problem_51 = undefined
```

## Problem 52

Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits in some order.

Solution:

```
problem_52 = undefined
```

## Problem 53

How many values of C(n,r), for 1 ≤ n ≤ 100, exceed one-million?

Solution:

```
problem_53 = undefined
```

## Problem 54

How many hands did player one win in the game of poker?

Solution:

```
problem_54 = undefined
```

## Problem 55

How many Lychrel numbers are there below ten-thousand?

Solution:

```
problem_55 = undefined
```

## Problem 56

Considering natural numbers of the form, ab, finding the maximum digital sum.

Solution:

```
problem_56 = undefined
```

## Problem 57

Investigate the expansion of the continued fraction for the square root of two.

Solution:

```
problem_57 = undefined
```

## Problem 58

Investigate the number of primes that lie on the diagonals of the spiral grid.

Solution:

```
problem_58 = undefined
```

## Problem 59

Using a brute force attack, can you decrypt the cipher using XOR encryption?

Solution:

```
problem_59 = undefined
```

## Problem 60

Find a set of five primes for which any two primes concatenate to produce another prime.

Solution:

```
problem_60 = undefined
```