Rules for Multiplying by 10, 100, and 1000 with Examples
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
| n | Multiple | Calculation |
|---|---|---|
| 1 | 42 | 42 × 1 |
| 2 | 84 | 42 × 2 |
| 3 | 126 | 42 × 3 |
| 4 | 168 | 42 × 4 |
| 5 | 210 | 42 × 5 |
| 6 | 252 | 42 × 6 |
| 7 | 294 | 42 × 7 |
| 8 | 336 | 42 × 8 |
| 9 | 378 | 42 × 9 |
| 10 | 420 | 42 × 10 |
| 11 | 462 | 42 × 11 |
| 12 | 504 | 42 × 12 |
| 13 | 546 | 42 × 13 |
| 14 | 588 | 42 × 14 |
| 15 | 630 | 42 × 15 |
| 16 | 672 | 42 × 16 |
| 17 | 714 | 42 × 17 |
| 18 | 756 | 42 × 18 |
| 19 | 798 | 42 × 19 |
| 20 | 840 | 42 × 20 |
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 12th multiple of 42.
- Write the formula: Multiple = \( 42 \times n \)
- Insert n = 12: \( 42 \times 12 = 504 \)
Answer: 504 is the 12th multiple of 42.
Example 2:
Is 336 a multiple of 42?
- Divide 336 by 42: \( 336 \div 42 = 8 \)
- 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 7th 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.
FAQs on How to Multiply by Multiples of 10 Easily
1. What is multiplying by multiples of 10?
Multiplying by multiples of 10 means multiplying a number by 10, 100, 1000, and so on. It's a fundamental skill in arithmetic, crucial for efficient calculations and understanding place value. This involves understanding the pattern of adding zeros when multiplying by powers of 10.
2. What is the rule for multiplying by 10, 100, or 1000?
The rule is simple: multiply the number by the non-zero part of the multiple of 10 and then add as many zeros as are in the multiple of 10. For example:
• 25 x 10 = 250 (add one zero)
• 25 x 100 = 2500 (add two zeros)
• 25 x 1000 = 25000 (add three zeros). This method relies on understanding place value and the properties of powers of ten.
3. What is an example of multiplying a number by a multiple of 10?
Here's an example: 12 x 30. You can break this down as (12 x 3) x 10 = 36 x 10 = 360. Alternatively, you can multiply 12 by 3 to get 36, then add a zero because you're multiplying by a multiple of 10 (30 has one zero).
4. How does place value help with multiplying by multiples of 10?
Place value is key! When you multiply by 10, each digit in the original number shifts one place to the left, increasing its value by a factor of 10. Multiplying by 100 shifts digits two places to the left, and so on. Understanding this shift makes mental math much easier.
5. Are there shortcut tricks for multiplying by multiples of 10 quickly?
Yes! The quickest method is to multiply the non-zero digits and then add the appropriate number of zeros to the end of the product. This is based on the fundamental properties of multiplication and place value.
6. What is the difference between multiplying by 10 and multiplying by 9 or 11?
Multiplying by 10 simply adds a zero to the end of the number. Multiplying by 9 or 11 requires a different approach; there isn't a simple 'add zeros' shortcut. Each requires different mental math strategies or algorithms.
7. Can you multiply decimals by multiples of 10 using the same rule?
Yes, but instead of adding zeros, you move the decimal point to the right. For every zero in the multiple of 10, move the decimal point one place to the right. This reflects the same underlying principle of place value.
8. Why does multiplying by multiples of 10 speed up mental maths?
It speeds up mental math because the process simplifies to adding zeros based on the number of zeros in the multiple. This eliminates the need for lengthy calculations and reduces potential errors. It relies on understanding place value and powers of ten.
9. Is there a pattern for multiplying any number by a two-digit multiple of 10 (like 40, 70, 90)?
Yes! First, multiply the number by the tens digit (e.g., 4 in 40). Then, multiply that result by 10 (add a zero). This combines the principles of distributive property and multiplication by powers of 10.
10. How is this skill important in algebra or advanced maths?
Understanding multiplication by multiples of 10 is crucial for understanding place value, powers of ten, and scientific notation. These concepts are foundational for algebra, calculus, and other advanced math topics.
11. What are some real-life applications of multiplying by multiples of 10?
Multiplying by multiples of 10 is used frequently in everyday life, for example: calculating the total cost of 10 items at $5 each, determining distances based on map scales, or even calculating time durations (10 minutes x 3 = 30 minutes).
12. How do I use multiplying by multiples of 10 to solve word problems?
When solving word problems, identify the key numbers and the relationship between them. If the problem involves groups of tens (or multiples of tens), this method is useful. Carefully read the problem and translate the words into a mathematical expression involving multiplication by multiples of 10 to get the answer.
