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The radical of \(n\), \({\rm rad}(n)\), is the product of distinct prime factors of \(n\). For example, \(504=2^3 \times 3^2 \times 7\), so \({\rm rad}(504)=2 \times 3 \times 7 = 42\). We shall define the triplet of positive integers \((a, \;b,\;c)\) to be an abc-hit if: 1. \({\rm GCD}(a, \;b)={\rm GCD}(a, \;c) = {\rm GCD}(b, \;c)=1\) 2. \(a

The minimum number of cubes to cover every visible face on a cuboid measuring \(3 \times 2 \times 1\) is twenty-two. If we then add a second layer to this solid it would require forty-six cubes to cover every visible face, the third layer would require seventy-eight cubes, and the fourth layer would require one hundred and eighteen cubes to cover every visible face. However, the first layer on a..

The palindromic number 595 is interesting because it can be written as the sum of consecutive squares: \(6^2 + 7^2 + 8^2 +9^2 + 10^2 + 11^2 + 12^2\). There are exactly eleven palindromes below one-thousand that can be written as consecutive square sums, and the sum of these palindromes is \(4164\). Note that \(1 = 0^2 + 1^2\) has not been included as this problem is concerned with the squares of..