Question

If 2 is the remainder when $m$ is divided by $5,$ what is the remainder when $3m$ is divided by $5?$

Short answer

The remainder is 1.

Step-by-step solution

- Understand the given condition

We are given that the remainder is 2 when the number 'm' is divided by 5. This can be mathematically expressed as: $$m\equiv 2(\mathrm{mod}5)$$

- Multiply both sides by 3

To find the remainder when $3m$ is divided by 5, multiply both sides of the congruence by 3: $$3m\equiv 3\cdot 2(\mathrm{mod}5)$$ which simplifies to: $$3m\equiv 6(\mathrm{mod}5)$$

- Simplify the expression

Now, simplify 6 modulo 5 to find the remainder. As 6 divided by 5 leaves a remainder of 1, we get: $$3m\equiv 1(\mathrm{mod}5)$$

Key concepts

Modular Arithmetic

Modular arithmetic is a system of arithmetic for integers, where numbers wrap around after reaching a certain value, known as the modulus. Think of it like a clock. Once you go past 12, you wrap around again to 1. The basic idea is to find the remainder when one number is divided by another.
For example, to show that 14 is the same as 2 under modulo 12, we write: 14 ≡ 2 (mod 12).
To write this mathematically: let m be a number and n be the modulus. The expression m ≡ r (mod n) means that the remainder when m is divided by n is r.
This notation helps simplify and solve certain types of mathematical problems, such as the one given in the exercise.

• One important property is that the same remainder operation applies to both sides of an equation if both sides are multiplied by the same number.
• This will be crucial as we solve the GMAT problem.

Understanding modular arithmetic allows you to handle equations and complex calculations more efficiently.

Remainder Theorem

The remainder theorem is a concept that ties polynomial division to evaluating functions. For our problem, understanding the simple principle of finding remainders can be very helpful. The remainder of a polynomial function when divided by a linear divisor gives the same result as plugging a particular value into the polynomial. However, for the number problems like the one in this GMAT exercise: 'When dealing with modular arithmetic, especially with integer division, it follows these steps:

• Take any number (say, m).
• Divide that number by another (like 5 in our problem).
• The remainder after division is your result under the modulus (r in m ≡ r (mod 5) ).

In our exercise, when m is divided by 5, the remainder is 2: m ≡ 2 (mod 5). When we multiply m by another number and find the modular result again, those rules still apply, as shown when we set up 3m ≡ ?. The ability to work with remainders systematically helps streamline problem-solving in modular arithmetic. It essentially breaks down complex operations into simpler, more manageable steps.

Multiplicative Properties

Multiplicative properties in the context of modular arithmetic highlight how multiplication affects remainders. Leveraging these properties can simplify calculations involving large numbers or variables.
For our given exercise:

• We started with a simple modular equation, m ≡ 2 (mod 5), meaning m leaves a remainder of 2 when divided by 5.
• To find what 3m leaves as a remainder, we multiply both sides by 3, following the multiplicative properties: 3m ≡ 6 (mod 5).

Since the remainder of 6 when divided by 5 is 1 (because 6=5*1+1), it simplifies to 3m ≡ 1 (mod 5).You can manipulate and solve modular equations by multiplying, simplifying within the modulus, and applying these properties consistently.
The consistency and logic make modular arithmetic a powerful tool for not just number problems but also algebraic expressions handled under specific modular systems.