REMAINDERS

This post is a part of [GMAT MATH BOOK]

Thanks to bb, Bunuel
I would add some new information and few questions that are testing these concepts in near future

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Definition

If \(x\) and \(y\) are positive integers, there exist unique integers \(q\) and \(r\), called the quotient and remainder, respectively, such that \(y =divisor*quotient+remainder= xq + r\) and \(0\leq{r}<x\).

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since \(15 = 6*2 + 3\).

Notice that \(0\leq{r}<x\) means that remainder is a non-negative integer and always less than divisor.

This formula can also be written as \(\frac{y}{x} = q + \frac{r}{x}\).

Properties

• When \(y\) is divided by \(x\) the remainder is 0 if \(y\) is a multiple of \(x\).
  For example, 12 divided by 3 yields the remainder of 0 since 12 is a multiple of 3 and \(12=3*4+0\).

• When a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer.
  For example, 7 divided by 11 has the quotient 0 and the remainder 7 since \(7=11*0+7\)

• The possible remainders when positive integer \(y\) is divided by positive integer \(x\) can range from 0 to \(x-1\).
  For example, possible remainders when positive integer \(y\) is divided by 5 can range from 0 (when y is a multiple of 5) to 4 (when y is one less than a multiple of 5).

• If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.
  For example, 123 divided by 10 has the remainder 3 and 123 divided by 100 has the remainder of 23.

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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Its a more complex problem when X is divide by two different int with two different remainder