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How many numbers between 1 and 520, including both, are divisible by 2 or 6?

Published in Number Theory 3 mins read

There are 346 numbers between 1 and 520 (inclusive) that are divisible by 2 or 6. Here's how we arrive at that answer:

Divisibility Rules and Concepts

  • Divisible by 2: A number is divisible by 2 if its last digit is even (0, 2, 4, 6, or 8).
  • Divisible by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
  • Inclusion-Exclusion Principle: To find the number of elements in the union of two sets (numbers divisible by 2 or 6), we add the number of elements in each set and then subtract the number of elements in their intersection. However, since every number divisible by 6 is already divisible by 2, finding numbers divisible by 6 provides no new elements to count.

Calculation Steps

  1. Numbers divisible by 2:

    • Divide the upper limit (520) by 2: 520 / 2 = 260. This means there are 260 numbers divisible by 2 between 1 and 520.

  2. Numbers divisible by 6:

    • Divide the upper limit (520) by 6: 520 / 6 = 86.666... We take the integer part: 86. This means there are 86 numbers divisible by 6 between 1 and 520.

  3. Considering Overlap: Because every number divisible by 6 is also divisible by 2, we don't need to use the inclusion-exclusion principle. All numbers divisible by 6 are already included in the count of numbers divisible by 2. However, to calculate the numbers divisible by 2 OR 6, and not divisible by both, it is found that the numbers divisible by 6 are part of the numbers divisible by 2, and don't need to be added again.

  4. Final Calculation:

    The question asks for the number of numbers divisible by 2 or 6. Since every number divisible by 6 is already divisible by 2, we need to count the numbers divisible by 2, PLUS those divisible by 6 that are not already divisible by 2. However, because all of the numbers that are divisible by 6 are also divisible by 2, we can simply add them all at once:

    Numbers divisible by 2 (260) + Numbers divisible by 6 (86) = 346

  5. To find the numbers only divisible by 2 you must subtract the amount of numbers that are divisible by 6 from the total numbers divisible by 2. In our problem that is, numbers divisible by 2 that are NOT divisible by 6 = 260 - 86 = 174. Add these numbers to the numbers divisible by 6 (86) and you find the total of numbers divisible by 2 or 6, 174+86=260. The key to solving this problem is to find the total amount of numbers divisible by the highest common factor, then the total amount divisible by one of the factors but not the highest common factor. Then add these two values together, and you will have your answer.

Conclusion

Therefore, there are 346 numbers between 1 and 520, inclusive, that are divisible by either 2 or 6.

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