Divisibility Multipliers
For each integer $p \gt 1$ coprime to $10$ there is a positive divisibility multiplier $m \lt p$ which preserves divisibility by $p$ for the following function...
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John Rhodes
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August 15, 2023
Sum of Squares
Consider equations of the form: $a^2 + b^2 = N$, $0 \le a \le b$, $a$, $b$ and $N$ integer. For $N=65$ there are two solutions: $a=1$,...
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John Rhodes
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August 15, 2023
Modular Cubes, Part 2
For a positive number $n$, define $C(n)$ as the number of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$. When...
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John Rhodes
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August 15, 2023
Modular Cubes, Part 1
For a positive number $n$, define $S(n)$ as the sum of the integers $x$, for which $1 \lt x \lt n$ and$x^3 \equiv 1 \bmod n$. When...
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John Rhodes
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August 15, 2023
Cutting Squares
A square piece of paper with integer dimensions $N \times N$ is placed with a corner at the origin and two of its sides along the $x$-...
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John Rhodes
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August 15, 2023
At Least Four Distinct Prime Factors Less Than 100
It can be verified that there are $23$ positive integers less than $1000$ that are divisible by at least four distinct primes less than $100$. Find how...
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John Rhodes
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August 15, 2023
Billionaire
You are given a unique investment opportunity. Starting with £1 of capital, you can choose a fixed proportion, f, of your capital to bet on a fair...
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John Rhodes
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August 15, 2023
Pseudo Square Root
The divisors of $12$ are: $1,2,3,4,6$ and $12$. The largest divisor of $12$ that does not exceed the square root of $12$ is $3$. We shall call...