To determine how many integers in the range from 1 to 140 are multiples of 2, 5, or 7, we can use the principle of inclusion-exclusion.
First, let’s calculate the number of multiples for each individual number:
- Multiples of 2: The largest multiple of 2 within 140 is 140. To find how many multiples of 2 there are, we divide 140 by 2:
140 / 2 = 70. So, there are 70 multiples of 2. - Multiples of 5: The largest multiple of 5 within 140 is 140. Dividing 140 by 5 gives:
140 / 5 = 28. Thus, there are 28 multiples of 5. - Multiples of 7: The largest multiple of 7 within 140 is also 140. Dividing 140 by 7 yields:
140 / 7 = 20. Hence, there are 20 multiples of 7.
Next, we need to account for the overlaps among these multiples:
- Multiples of 10 (which is 2 and 5): The largest multiple of 10 within 140 is 140. We have:
140 / 10 = 14. So there are 14 multiples of 10. - Multiples of 14 (which is 2 and 7): The largest multiple of 14 within 140 is 140. Therefore:
140 / 14 = 10, so there are 10 multiples of 14. - Multiples of 35 (which is 5 and 7): The largest multiple of 35 within 140 is 140. Thus:
140 / 35 = 4, giving us 4 multiples of 35.
Finally, we need to consider multiples of 70 (which is 2, 5, and 7): The largest multiple of 70 within 140 is 140. Hence, we have: 140 / 70 = 2, so there are 2 multiples of 70.
Now, applying the principle of inclusion-exclusion:
Number of multiples of 2, 5, or 7 = (Multiples of 2) + (Multiples of 5) + (Multiples of 7) - (Multiples of 10) - (Multiples of 14) - (Multiples of 35) + (Multiples of 70)
Substituting the values we obtained:
70 + 28 + 20 - 14 - 10 - 4 + 2 = 92
Therefore, the total number of integers between 1 and 140 that are multiples of 2, 5, or 7 is 92.