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EASY-MEDIUM, MATHS
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Coprime,multiplication,Logic
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given positive integers x1, x2, . . . , xn with gcd(x1, x2, . . . , xn) = 1, compute the largest integer not representable as a non-negative integer linear combination of the xi . This largest integer is sometimes denoted g(x1, . . . , xn). The restriction gcd(x1, x2, . . . , xn) = 1 is necessary for the definition to be meaningful, for otherwise every non-negative integer linear combination is divisible by this gcd in that case answer would be infinity
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Given 2 numbers 'a' and 'b' there equation is ax+by,a and b are coprime
So all number can be generate just plot ax+by=N find a and and but the condition here was that a and b cannot be negative,in that case only will there be a maximum number that cannot be generated. consider 7,9 7x+9y=k x=(k-9y)/7
a*(b-1)+b*-1=a*b-a-b In the above table we see that with multiples of 9 we have generated all possible remainders of 7. so x=(k-9y)/7 here x can always be intere for any value of k because whatever value of k we take it will give a remainder 0 to 6 and we will put y accordingly to compensate that remainder like k=101 k%7=3 so put y=5 x=(101-9*5)/7 =8 |
PROBLEM LINK:DIFFICULTY:
EASY-MEDIUM, MATHS
PREREQUISITES:
Division, Summation of series, modular operation
PROBLEM:
Given a number N and another number MOD, you have to calculate the sum of numbers from 1 to N as:
EXPLANATION:
By the division algorithm, let N = qm+r with 0 ≤ r < m. Consider the sequence (1 % m),(2 % m), ...,(N % m). It looks like this:
There are q layers, each with m elements and a final layer with r elements.
We know that sum of first N terms is
Therefore using this you can easily calculate sum of large values of N also is O(1) complexity.
AUTHOR'S AND TESTER'S SOLUTIONS:
Solution can be found here.
Mr IDC and his Message - Editorial
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