Multiplying By Multiples Of 10 Explained With Simple Steps

The concept of Multiples of 42 is an essential foundation in number theory and arithmetic that students encounter in school maths and competitive exams like JEE and NEET. Knowing how to find and use multiples of 42 is valuable for understanding patterns, division, and solving many higher-level maths problems.

Understanding Multiples of 42

A multiple of 42 is any number that can be expressed as \( 42 \times n \), where \( n \) is an integer (whole number). This means you can create multiples by multiplying 42 by 1, 2, 3, 4, and so on. For example, the first few multiples of 42 are 42, 84, 126, and 168. Recognizing multiples is useful for solving division, factors, and multiples problems in school and Olympiad competitions.

How to Find Multiples of 42

To find multiples of 42, simply multiply 42 by natural numbers:

• First multiple: \( 42 \times 1 = 42 \)
• Second multiple: \( 42 \times 2 = 84 \)
• Third multiple: \( 42 \times 3 = 126 \)
• And so on: \( 42 \times 4 = 168 \), \( 42 \times 5 = 210 \), ...

You can also get each multiple by repeatedly adding 42 to the previous number. For example: 42 + 42 = 84, 84 + 42 = 126, and so on. This approach emphasizes the connection to repeated addition, a key building block for understanding multiplication.

First 20 Multiples of 42

Properties and Related Concepts

• Every multiple of 42 ends with a digit from the multiples of 2, 3, and 7, because 2 × 3 × 7 = 42 (prime factorization).
• All multiples of 42 are even numbers, since 42 itself is even.
• If a number is a multiple of 42, it is also a multiple of 2, 3, and 7.
• Multiples of 42 are helpful in finding the LCM (lowest common multiple) with other numbers.
• A multiple common to two numbers (for example, 42 and 30) is called a common multiple.

Worked Examples

Example 1:

Find the 12^th multiple of 42.

1. Write the formula: Multiple = \( 42 \times n \)
2. Insert n = 12: \( 42 \times 12 = 504 \)

Answer: 504 is the 12th multiple of 42.

Example 2:

Is 336 a multiple of 42?

1. Divide 336 by 42: \( 336 \div 42 = 8 \)
2. Since the result is a whole number, 336 is a multiple of 42.

Answer: Yes, 336 is a multiple of 42 (since 42 × 8 = 336).

Example 3:

List all common multiples of 14 and 42 up to 200.

• Multiples of 14 up to 200: 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196
• Multiples of 42 up to 200: 42, 84, 126, 168
• Common multiples: 42, 84, 126, 168

Answer: 42, 84, 126, 168

Practice Problems

• Find the 7^th multiple of 42.
• Which is the least common multiple of 42 and 30?
• List all multiples of 42 between 400 and 600.
• Is 225 a multiple of 42? Justify your answer.
• Write the next three multiples after 672 for 42.

Common Mistakes to Avoid

• Confusing multiples with factors—remember, multiples are results of multiplication, while factors divide the number exactly.
• Forgetting to use only natural numbers (1, 2, 3, …) when finding multiples.
• Assuming multiples are only less than the number—multiples always start from the number itself and go higher.

Real-World Applications

Multiples of 42 appear in daily life when grouping or packaging things. For example, if a shopkeeper puts 42 candies into a box, he can easily count out 84, 126, or 168 candies using the multiples. In music, rhythm patterns may also use 42-beat cycles. Understanding multiples also helps with time calculations, dividing task loads, and organizing events at regular intervals.

Page Summary

This page has explored multiples of 42, explained how to find them, provided worked examples, and highlighted real-world uses. Mastering this basic arithmetic concept is crucial for faster calculations, advanced topics, and competitive exams. At Vedantu, we make learning numbers and patterns fun and easy by using plenty of worked examples and real-world contexts. To deepen your maths skills, check out related topics like Multiples of 4 and Factors of 42, or learn more about Division on Vedantu.