I recently encountered this form of representation of integer numbers: $a_j=2^{k_j}q_j$, where $k_j$ is a non negative integer and $q_j$ is odd. My question is from where one can get idea of representing integers in this kind of forms?
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You just factor out as many 2's as possible. Once you can't factor out any more, the result must be odd. – Jair Taylor Aug 06 '17 at 05:36
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This representation comes from the Fundamental Theorem of Arithmetic, and changes a nonzero integer into a unique pair of integers, one nonnegative and the other odd. We can do more:
$$a_j=2^{k_j}3^{i_j}q_j$$ Now, $k_j$ and $i_j$ are nonnegative integers, and $q_j$ is a nonzero integer having no divisors in common with $6$. This representation is also unique, now as a triple.
There's nothing special about $2$, or $2$ and $3$; any primes can be used in this manner. All that matters is that the leftover bit, the $q_j$, has none of those primes dividing it.
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