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코딩 연습
Given the positive integers \(x,\; y\), and \(z\), are consecutive terms of an arithmetic progression, the least value of the positive integers \(n\), for which the equation, \(x^2 - y^2 - z^2 =n\), has exactly two solutions is \(n=27\):\[34^2-27^2-20^2=12^2-9^2-6^2=27\] It turns out that \(n=1155\) is the least value which has exactly ten solutions. How many values of \(n\) less than one millio..
Consider the consecutive primes \(p_1=19\) and \(p_2=23\). It can be verified that \(1219\) is the smallest number such that the last digits are formed by \(p_1\) whilst also being divisible by \(p_2\). In fact, with the exception of \(p_1-3\) and \(p_2=5\), for every pair of consecutive primes, \(p_2 > p_1\), there exist values of \(n\) for which the last digits are formed by \(p_1\) and \(n\) ..
A number consisting entirely of ones is called a repunit. We shall define \(R(k)\) to be a repunit of length \(k\); for example, \(R(6)=111111\). Let us consider repunits of the form \(R \left (10^n \right )\). Although \(R(10),\; R(100)\), or \(R(1000)\) are not divisible by \(17\), \(R(10000)\) is divisible by \(17\). Yet there is no value of \(n\) for which \(R \left ( 10^n \right )\) will di..