What is the modular representation of an integer? What is the modular representation of an integer using a set of primes?
More specifically, a problem on my homework asks to convert 49 to a modular representation using primes 7,11,13,17.
Would appreciate a general solution.
 A: Your textbook surely states what they mean by the phrase.
But to take a guess, they're probably referring to a "residue number system", where you represent not-too-large integers as a sequence of residue classes modulo a set of moduli (usually primes).
(to do the reverse conversion, from the residue number system to decimal (or other representations) typically involves the Chinese Remainder Theorem)
A: I saw this in the book, Applied Algebra- Codes, Ciphers and Discrete Logarithms by Darel W.Hardy. It's on page 138 of the second edition, in the section about the Chinese Remainder Theorem and Extended Precision Arithmetic. The accepted answer gives a wonderful summary of the book's content.
In the general case, the author states, "Let $m_1,m_2,...,m_r$" be pairwise relatively prime positive integers with the property that $m_i^2$ can be computed exactly in machine arithmetic. Then any positive integer $n$ less than the product $m_1m_2...m_r$ is determined by the numbers $$ a_1  = n\: mod\:m_1, a_2  = n\: mod\:m_2,...,a_r  = n\: mod\:m_r$$ The modular representation of n relative to the bases $m_1,m_2,...,m_r$ takes the form $$ n = [a_1, a_2,...,a_r]$$
Knowing this, we can represent 49 using the primes 7,11,13,17.
$$49\:mod\:7 = 0$$ $$49\:mod\:11 = 5$$ $$49\:mod\:13 = 10$$ $$49\:mod\:17 = 15$$ We get $$ 49 = [0, 5,10,15]$$
We can get 49 from its modular representation by solving the system of congruences
$$x \equiv 0\;mod\: 7$$ $$x \equiv 5\;mod\: 11$$ $$x \equiv 10\;mod\: 13$$ $$x \equiv 15\;mod\: 17$$
