Combinatoric selections

There are exactly ten ways of selecting three from five, 12345:

```        123, 124, 125, 134, 135, 145, 234, 235, 245, and 345
```

In combinatorics, we use the notation, C(3,5) = 10.

Permuted multiples

It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.

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

Prime digit replacements

By replacing the 1st digit of the 2-digit number *3, it turns out that six of the nine possible values: 13, 23, 43, 53, 73, and 83, are all prime.

By replacing the 3rd and 4th digits of 56**3 with the same digit, this 5-digit...

»

Prime permutations

The arithmetic sequence, 1487, 4817, 8147, in which each of the terms increases by 3330, is unusual in two ways: (i) each of the three terms are prime, and, (ii) each of the 4-digit numbers are permutations of one another.

There are no arithmetic sequences made up of three 1-, 2-, or 3-digit primes, exhibiting this property, but there is one other 4-digit increasing sequence.

Self powers

The series, 1^1 + 2^2 + 3^3 + … + 10^10 = 10405071317.

Find the last ten digits of the series, 1^1 + 2^2 + 3^3 + … + 1000^1000.

Distinct primes factors

The first two consecutive numbers to have two distinct prime factors are:

```        14 = 2 × 7
15 = 3 × 5
```

The first three consecutive numbers to have three distinct prime factors are:

```        644 = 2² × 7 × 23
645 = 3 × 5 × 43
646 = 2 × 17 × 19.
```