What are the Multiples of 35?

The multiples of 35 are numbers that can be obtained by multiplying 35 by any positive integer. To find the multiples of 35, we can start by listing the product of 35 multiplied by the integers 1, 2, 3, and so on. Let's explore the multiples of 35 up to a certain point.

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35 × 1 = 35 35 × 2 = 70 35 × 3 = 105 35 × 4 = 140 35 × 5 = 175 35 × 6 = 210 35 × 7 = 245 35 × 8 = 280 35 × 9 = 315 35 × 10 = 350

These are the first ten multiples of 35. As we can observe, each multiple is obtained by multiplying 35 by an integer. The pattern continues indefinitely, and we can generate more multiples by continuing the multiplication process.

The general formula to express the multiples of 35 is:

35 × n, where n is any positive integer.

So, the multiples of 35 are an infinite set of numbers that follow this pattern. They include 35 itself and extend infinitely as we multiply 35 by 2, 3, 4, and so on. These multiples find applications in various mathematical and real-world contexts, such as in arithmetic, algebraic expressions, and problem-solving situations where multiples play a crucial role.

What is a Multiple?

In mathematics, the term "multiple" refers to the product of any quantity and an integer. More specifically, if you have a whole number (an integer) and you multiply it by another integer, the result is a multiple of the original number.

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For example, let's consider the number 3. Multiples of 3 would be obtained by multiplying 3 by any integer. So, the first few multiples of 3 are:

  • 3×1=3
  • 3×2=6
  • 3×3=9
  • 3×4=12

And so on. The set of all multiples of 3 would include numbers like 3, 6, 9, 12, 15, and so forth.

In general, if a and b are integers, then b is a multiple of a if there exists an integer c such that b=a×c.

First 20 Multiples of 35

To find the first 20 multiples of 35, you can multiply 35 by each integer from 1 to 20. Here they are:

  1. 35
  2. 70
  3. 105
  4. 140
  5. 175
  6. 210
  7. 245
  8. 280
  9. 315
  10. 350
  11. 385
  12. 420
  13. 455
  14. 490
  15. 525
  16. 560
  17. 595
  18. 630
  19. 665
  20. 700

Each number in the list is a multiple of 35.

How to Find the Multiples of 35?

To find the multiples of 35, you can follow these steps:

  1. List the Multiples:

    • Start by writing down the first few natural numbers (1, 2, 3, 4, ...).
    • Multiply each natural number by 35.

  2. Identify Multiples:

    • Look for numbers that result from the multiplication of 35.

For example:

  • 1×35=35
  • 2×35=70
  • 3×35=1053
  • 4×35=140
  • 5×35=175

So, the multiples of 35 are 35, 70, 105, 140, 175, and so on. You can continue this process to find more multiples by multiplying consecutive natural numbers by 35.

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Solved Examples on the Multiples of 35

Here are some solved examples involving multiples of 35:

Example 1: Finding the first 10 multiples of 35

We know that a multiple of 35 is any number that leaves no remainder when divided by 35. So, the first few multiples of 35 are:

35, 70, 105, 140, 175, 210, 245, 280, 315, 350

Therefore, the first 10 multiples of 35 are:

35, 70, 105, 140, 175, 210, 245, 280, 315, 350

Example 2: Checking if a number is a multiple of 35

Let's check if the number 245 is a multiple of 35. We can do this by dividing 245 by 35:

245 ÷ 35 = 7 (with no remainder)

Since there is no remainder, we can confirm that 245 is a multiple of 35.

Example 3: Finding the least common multiple (LCM) of 35 and 70

The LCM of two numbers is the smallest number that is a multiple of both numbers. To find the LCM of 35 and 70, we can:

  1. List out the multiples of 35 and 70:
    • Multiples of 35: 35, 70, 105, 140, 175, 210, ...
    • Multiples of 70: 70, 140, 210, 280, 350, ...
  2. Identify the first number that appears in both lists: 350
  3. Therefore, the LCM of 35 and 70 is 350.

Example 4: Finding the next multiple of 35 after a given number

Let's find the next multiple of 35 after 280. We can do this by:

  1. Dividing 280 by 35: 280 ÷ 35 = 8
  2. Adding 1 to the quotient: 8 + 1 = 9
  3. Multiplying the result by 35: 9 × 35 = 315

Therefore, the next multiple of 35 after 280 is 315.

These are just a few examples of how to work with multiples of 35. Remember that you can always use the definition of a multiple (a number that leaves no remainder when divided by another number) to solve any problem involving multiples.

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