Maximum Product of Parts
Let $N$ be a positive integer and let $N$ be split into $k$ equal parts, $r = N/k$, so that $N = r + r + \cdots...
-
John Rhodes
-
August 15, 2023
RSA Encryption
The RSA encryption is based on the following procedure: Generate two distinct primes $p$ and $q$.Compute $n = pq$ and $\phi = (p – 1)(q – 1)$....
-
John Rhodes
-
August 15, 2023
Grouping Two Different Coloured Objects
Having three black objects B and one white object W they can be grouped in 7 ways like this: (BBBW)(B,BBW)(B,B,BW)(B,B,B,W) (B,BB,W)(BBB,W)(BB,BW) In how many ways can sixty...
-
John Rhodes
-
August 15, 2023
Golden Triplets
For any integer $n$, consider the three functions \begin{align} f_{1, n}(x, y, z) &= x^{n + 1} + y^{n + 1} – z^{n + 1}\\ f_{2, n}(x,...
-
John Rhodes
-
August 15, 2023
Consecutive Positive Divisors
Find the number of integers $1 \lt n \lt 10^7$, for which $n$ and $n + 1$ have the same number of positive divisors. For example, $14$...
-
John Rhodes
-
August 15, 2023
Step Numbers
Consider the number $45656$. It can be seen that each pair of consecutive digits of $45656$ has a difference of one. A number for which every pair...
-
John Rhodes
-
August 15, 2023
Integer Angled Quadrilaterals
Let $ABCD$ be a convex quadrilateral, with diagonals $AC$ and $BD$. At each vertex the diagonal makes an angle with each of the two sides, creating eight...
-
John Rhodes
-
August 15, 2023
Common Cathetus Right-angled Triangles
The four right-angled triangles with sides $(9,12,15)$, $(12,16,20)$, $(5,12,13)$ and $(12,35,37)$ all have one of the shorter sides (catheti) equal to $12$. It can be shown that...