HOW MANY POSITIVE INTEGERS LESS THAN 1000
A.are divisible by exactly one of 7 and 11?
B. are divisible by neither 7 nor 11?
C. have distinct digits?
D. have distinct digits and are even?
To find the number of positive integers that satisfy each condition, we can use basic principles of counting and apply them to each case.
A. To find the numbers that are divisible by exactly one of 7 and 11, we can use the principle of inclusion-exclusion. First, we find the numbers divisible by 7 by dividing 1000 by 7 (since we are looking for integers less than 1000), which gives us 142. Then, we find the numbers divisible by 11 using the same method, which gives us 90. However, we need to subtract the numbers that are divisible by both 7 and 11, as they were counted twice. To do that, we find the numbers divisible by 7 and 11 simultaneously by dividing 1000 by their least common multiple, which is 77. There are 12 numbers divisible by both 7 and 11. So, the numbers divisible by exactly one of 7 and 11 is 142 + 90 - 12 = 220.
B. To find the numbers that are divisible by neither 7 nor 11, we need to subtract the numbers that are divisible by 7 or 11 or both from the total count of positive integers less than 1000. The count of numbers divisible by 7 is 142, the count of numbers divisible by 11 is 90, and the count of numbers divisible by both 7 and 11 is 12. So, the count of numbers divisible by either 7 or 11 is 142 + 90 - 12 = 220. Therefore, the count of numbers divisible by neither 7 nor 11 is 1000 - 220 = 780.
C. To find the numbers that have distinct digits, we can count the possibilities for each digit separately. For the first digit, we have 9 choices (1 to 9). For the second digit, we have 9 choices (0 to 9 excluding the digit already chosen for the first digit). For the third digit, we have 8 choices (0 to 9 excluding the two digits already chosen). So, the total count of numbers with distinct digits is 9 * 9 * 8 = 648.
D. To find the numbers that have distinct digits and are even, we need to further restrict the possibilities for the last digit to be even. For the first two digits, we can still choose from 9 choices for each. However, for the last digit, we have 4 choices (0, 2, 4, 6). So, the count of numbers with distinct digits and even is 9 * 9 * 4 = 324.
In summary:
A. There are 220 positive integers less than 1000 that are divisible by exactly one of 7 and 11.
B. There are 780 positive integers less than 1000 that are divisible by neither 7 nor 11.
C. There are 648 positive integers less than 1000 that have distinct digits.
D. There are 324 positive integers less than 1000 that have distinct digits and are even.