To find how many positive integers less than 1000 are divisible by both 7 and 11, we first need to determine the least common multiple (LCM) of these two numbers. The LCM of 7 and 11 can be calculated because both are prime numbers. Hence, the LCM is simply their product:
LCM(7, 11) = 7 * 11 = 77
Next, we need to find how many multiples of 77 are less than 1000. We can find this by dividing 1000 by 77 and rounding down to the nearest whole number since we need only positive integers.
Calculating:
1000 / 77 ≈ 12.99
Rounding down gives us 12. Therefore, there are 12 positive integers less than 1000 that are divisible by both 7 and 11.
To summarize, the count of positive integers less than 1000 that are divisible by both 7 and 11 is 12.