Ordered Radicals
The radical of $n$, $\operatorname{rad}(n)$, is the product of the distinct prime factors of $n$. For example, $504 = 2^3 \times 3^2 \times 7$, so $\operatorname{rad}(504) =...
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John Rhodes
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August 15, 2023
Prime Square Remainders
Let $p_n$ be the $n$th prime: $2, 3, 5, 7, 11, \dots$, and let $r$ be the remainder when $(p_n – 1)^n + (p_n + 1)^n$ is...
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John Rhodes
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August 15, 2023
Efficient Exponentiation
The most naive way of computing $n^{15}$ requires fourteen multiplications: $$n \times n \times \cdots \times n = n^{15}.$$ But using a “binary” method you can compute...
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John Rhodes
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August 15, 2023
Disc Game Prize Fund
A bag contains one red disc and one blue disc. In a game of chance a player takes a disc at random and its colour is noted....
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John Rhodes
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August 15, 2023
Square Remainders
Let $r$ be the remainder when $(a – 1)^n + (a + 1)^n$ is divided by $a^2$. For example, if $a = 7$ and $n = 3$,...
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John Rhodes
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August 15, 2023
Digit Power Sum
The number $512$ is interesting because it is equal to the sum of its digits raised to some power: $5 + 1 + 2 = 8$, and...
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John Rhodes
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August 15, 2023
Pandigital Prime Sets
Using all of the digits $1$ through $9$ and concatenating them freely to form decimal integers, different sets can be formed. Interestingly with the set $\{2,5,47,89,631\}$, all...
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John Rhodes
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August 15, 2023
Red, Green, and Blue Tiles
Using a combination of grey square tiles and oblong tiles chosen from: red tiles (measuring two units), green tiles (measuring three units), and blue tiles (measuring four...