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2^{2}\times 5^{2}\times 7\times 11
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factor(7700)
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The number of ordered triples (a, b, c) of positive integers which satisfy the simultaneous equations ab + bc = 44, ac + bc = 33
https://math.stackexchange.com/q/664860
Your solution is correct. Noe that a=1,b-c=11 and a=11, b-c=1 both lead to a+b=12+c. Then 33=ac+bc=(12+c)c indeed has no solution. [There's an alomost-soution: c=-1 gives -33; so I wonder ...
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Note that the period for a prime p is a factor of \varphi (p)=p-1 but need not be equal to it. This is because 10^{p-1} \equiv 1 \mod p. The period is the least n for which p|(10^n-1). If ...
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Write the first constraint as \sum_j a_{1j} = m, with m < n ( I am just replacing the variable x by m). Since a_{i1}+a_{i2} = 1, and a_{ij}\geq 0, we see that this is equivalent to a_{i2} = 1-a_{i1} ...
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