To find the number of positive integers less than 1000 that are divisible by both 7 and 11, we first need to determine their least common multiple (LCM).
The LCM of 7 and 11 can be calculated since both are prime numbers. Therefore, the LCM is simply:
LCM(7, 11) = 7 * 11 = 77
Now, we need to find how many multiples of 77 are there that are less than 1000. We can do this by dividing 1000 by 77:
1000 ÷ 77 ≈ 12.99
Since we are interested in positive integers, we take the integer part of the division, which gives us 12. This means there are 12 full multiples of 77 that are less than 1000.
To confirm, we can list the multiples of 77 that are less than 1000:
- 77 x 1 = 77
- 77 x 2 = 154
- 77 x 3 = 231
- 77 x 4 = 308
- 77 x 5 = 385
- 77 x 6 = 462
- 77 x 7 = 539
- 77 x 8 = 616
- 77 x 9 = 693
- 77 x 10 = 770
- 77 x 11 = 847
- 77 x 12 = 924
All these values are indeed less than 1000. Therefore, the final answer is:
There are 12 positive integers less than 1000 that are divisible by both 7 and 11.