To find how many integers in the range 1 through 140 are multiples of 2, 5, or 7, we can use the principle of inclusion-exclusion.
First, we calculate the number of multiples for each individual integer:
- Multiples of 2: The largest multiple of 2 in this range is 140, and the smallest is 2. The multiples of 2 form the sequence: 2, 4, 6, …, 140. This is an arithmetic series where the first term (a) is 2 and the last term (l) is 140, with a common difference (d) of 2. To find the number of terms (n), we can use the formula:
n = (l – a) / d + 1 = (140 – 2) / 2 + 1 = 70. - Multiples of 5: The largest multiple of 5 in this range is 140, and the smallest is 5. The multiples of 5 form the sequence: 5, 10, 15, …, 140. Using the same formula:
n = (140 – 5) / 5 + 1 = 28. - Multiples of 7: The largest multiple of 7 in this range is 140, and the smallest is 7. The multiples of 7 form the sequence: 7, 14, 21, …, 140. Using the formula:
n = (140 – 7) / 7 + 1 = 20.
Now we apply inclusion-exclusion to avoid double-counting:
- Multiples of both 2 and 5 (i.e., multiples of 10):
n = (140 – 10) / 10 + 1 = 14. - Multiples of both 2 and 7 (i.e., multiples of 14):
n = (140 – 14) / 14 + 1 = 10. - Multiples of both 5 and 7 (i.e., multiples of 35):
n = (140 – 35) / 35 + 1 = 4. - Multiples of 2, 5, and 7 (i.e., multiples of 70):
n = (140 – 70) / 70 + 1 = 2.
Putting it all together, we have:
Total = (Multiples of 2) + (Multiples of 5) + (Multiples of 7)
- (Multiples of 10) - (Multiples of 14) - (Multiples of 35)
+ (Multiples of 70)
Total = 70 + 28 + 20 - 14 - 10 - 4 + 2 = 92.
So, there are a total of 92 integers in the range from 1 to 140 that are multiples of 2, 5, or 7.