Welcome to the foundational module on Integers, Divisibility, and Remainders. These concepts are crucial building blocks for many quantitative reasoning sections in competitive exams like the GMAT. Mastering them will unlock your ability to solve a wide range of problems efficiently.

Integers are whole numbers, including positive numbers, negative numbers, and zero. They do not have fractional or decimal parts. The set of integers is denoted by $\mathbb{Z}$ and includes ..., -3, -2, -1, 0, 1, 2, 3, ...

An integer 'a' is said to be divisible by another integer 'b' (where b is not zero) if the division of 'a' by 'b' results in an integer with no remainder. In other words, if $a = b \times k$ for some integer $k$, then 'a' is divisible by 'b'. We often say 'b' is a factor or divisor of 'a', and 'a' is a multiple of 'b'.

For example, 12 is divisible by 3 because $12 = 3 \times 4$. Here, 3 is a factor of 12, and 12 is a multiple of 3. However, 12 is not divisible by 5 because there's no integer $k$ such that $12 = 5 \times k$.

Knowing divisibility rules can significantly speed up problem-solving. Here are some common ones:

When an integer 'a' is divided by a non-zero integer 'b', we get a quotient 'q' and a remainder 'r'. This can be expressed as: $a = bq + r$, where $0 \le r < |b|$. The remainder 'r' is the amount 'left over' after dividing 'a' into as many whole groups of 'b' as possible.

Important properties of remainders:

• The remainder is always non-negative and strictly less than the absolute value of the divisor.

• If a number is divisible by another, the remainder is 0.

Let's apply these concepts. Consider the number 345. Is it divisible by 3? Yes, because $3+4+5 = 12$, and 12 is divisible by 3. What is the remainder when 345 is divided by 5? It ends in 5, so it's divisible by 5, meaning the remainder is 0.

Understanding the relationship between integers, divisibility, and remainders is fundamental. Practice these concepts regularly to build confidence and speed for your exams.