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...
-
John Rhodes
-
August 15, 2023
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,...
-
John Rhodes
-
August 15, 2023
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,...
-
John Rhodes
-
August 15, 2023
Digit Fifth Powers
Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits: \begin{align} 1634 &= 1^4 + 6^4 + 3^4...
-
John Rhodes
-
August 15, 2023
Consecutive Prime Sum
The prime $41$, can be written as the sum of six consecutive primes: $$41 = 2 + 3 + 5 + 7 + 11 + 13.$$ This...
-
John Rhodes
-
August 15, 2023
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...
-
John Rhodes
-
August 15, 2023
Self Powers
The series, $1^1 + 2^2 + 3^3 + \cdots + 10^{10} = 10405071317$. Find the last ten digits of the series, $1^1 + 2^2 + 3^3 +...
-
John Rhodes
-
August 15, 2023
Distinct Primes Factors
The first two consecutive numbers to have two distinct prime factors are: \begin{align} 14 &= 2 \times 7\\ 15 &= 3 \times 5. \end{align} The first three...