Chapter 14: Problem 10
How many positive integers less than 50 are multiples of 4 but NOT multiples of \(6 ?\)
Short Answer
Expert verified
There are 8 numbers.
Step by step solution
01
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.
02
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.
03
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.
04
Count the remaining numbers
Lastly, count the remaining numbers. There are 8 numbers: 4, 8, 16, 20, 28, 32, 36, and 44.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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.
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.
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.
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:
Practice identifying and generating number sequences to improve your math problem-solving skills.
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.
Practice identifying and generating number sequences to improve your math problem-solving skills.