Question

How many positive integers less than 50 are multiples of 4 but NOT multiples of $6?$

Short answer

There are 8 numbers.

Step-by-step solution

Identify multiples of 4

First, find the positive integers less than 50 that are multiples of 4. These can be found by calculating the sequence: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, and 48.

Identify multiples of 6

Next, identify the multiples of 4 that are also multiples of 6. The least common multiple (LCM) of 4 and 6 is 12. So, we need to find multiples of 12 that are also less than 50 within the list identified in Step 1. These numbers are 12, 24, and 48.

Subtract multiples of 12

Now, remove the identified multiples of 12 from the list of multiples of 4. The remaining numbers are 4, 8, 16, 20, 28, 32, 36, and 44.

Count the remaining numbers

Lastly, count the remaining numbers. There are 8 numbers: 4, 8, 16, 20, 28, 32, 36, and 44.

Key concepts

Multiples

In math, a multiple is the product of a number and an integer. For example, 4, 8, and 12 are multiples of 4 because they result from multiplying 4 by 1, 2, and 3, respectively.
To identify multiples of a number, list the products of the number with successive integers:

• For 4: 4, 8, 12, 16, 20, and so on.
• For 6: 6, 12, 18, 24, 30, etc.

In problems like the given exercise, finding multiples can help identify numbers that fit specific criteria.

Integers

Integers are whole numbers and include positive numbers, negative numbers, and zero. For example, -3, 0, and 7 are all integers. They do not have fractional or decimal components.
When solving GMAT math problems like our exercise, focus on positive integers if specified. Here, we identified positive integers that are multiples of 4 but not multiples of 6. Always ensure you understand whether the problem is asking for positive, negative, or both types of integers to provide an accurate solution.

Least Common Multiple (LCM)

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. For example, the LCM of 4 and 6 is 12 because it's the smallest number divisible by both 4 and 6.
To find the LCM, you can use the prime factorization method or list the multiples of each number and find the smallest common value.

• Prime Factorization: Break numbers into prime factors.
• Identify the highest power of each prime factor and multiply them.

Find the multiples of the LCM to solve problems where the LCM plays a role, such as identifying numbers that are multiples of both 4 and 6.

Number Sequences

Number sequences are ordered lists of numbers following a specific pattern. They can be arithmetic (adding a constant value) or geometric (multiplying by a constant value).
In our problem:

• First, we identified the sequence of multiples of 4.
• Next, we found a subset of this sequence that are also multiples of 6, forming another sequence with a step of 12.

By using these sequences, students can visualize and solve complex problems more easily.
Practice identifying and generating number sequences to improve your math problem-solving skills.