Consider all integers from 1 up to and including 300. Find the number of them that are divisible by:(a) at least one of 3, 5, 7;(c) by 5, but by neither 3 nor 7;(b) 3 and 5 but not by 7;(d) by none of the numbers 3, 5, 7.
To solve these problems, we can use the concept of divisibility rules and basic counting principles. Let's solve each part step by step:
(a) Find the number of integers divisible by at least one of 3, 5, or 7:
To find the number of integers divisible by at least one of 3, 5, or 7, we need to find the union of the sets of numbers divisible by 3, 5, and 7. We can apply the inclusion-exclusion principle.
Step 1: Calculate the number of integers divisible by 3:
There are a total of 300/3 = 100 integers divisible by 3.
Step 2: Calculate the number of integers divisible by 5:
There are 300/5 = 60 integers divisible by 5.
Step 3: Calculate the number of integers divisible by 7:
There are 300/7 = 42 integers divisible by 7.
To find the union of these sets, we need to add the numbers divisible by 3, 5, and 7, then subtract the numbers divisible by both pairs (3 and 5, 3 and 7, 5 and 7), then add the numbers divisible by 3, 5, and 7.
Number of integers divisible by 3: 100
Number of integers divisible by 5: 60
Number of integers divisible by 7: 42
Number of integers divisible by both 3 and 5: 300/15 = 20
Number of integers divisible by both 3 and 7: 300/21 = 14
Number of integers divisible by both 5 and 7: 300/35 = 8
Number of integers divisible by all three (3, 5, and 7): 300/105 = 2
Now, we can calculate the number of integers divisible by at least one of 3, 5, or 7:
100 + 60 + 42 - 20 - 14 - 8 + 2 = 162
So, the number of integers divisible by at least one of 3, 5, or 7 is 162.
(b) Find the number of integers divisible by 3 and 5 but not by 7:
To find the number of integers divisible by 3 and 5 but not by 7, we need to find the intersection of the sets of numbers divisible by 3 and 5, then subtract the numbers divisible by 7.
Number of integers divisible by both 3 and 5: 300/15 = 20
Number of integers divisible by 7: 300/7 = 42
Number of integers divisible by both 3, 5, and 7: 2
Number of integers divisible by 3 and 5 but not 7: 20 - 2 = 18
So, the number of integers divisible by 3 and 5 but not by 7 is 18.
(c) Find the number of integers divisible by 5 but not by 3 or 7:
To find the number of integers divisible by 5 but not by 3 or 7, we need to find the set of numbers divisible by 5 and then subtract the numbers divisible by both 3 and 5 or 7.
Number of integers divisible by 5: 300/5 = 60
Number of integers divisible by both 3 and 5: 20
Number of integers divisible by both 5 and 7: 8
Number of integers divisible by 5 but not by 3 or 7: 60 - 20 - 8 = 32
So, the number of integers divisible by 5 but not by 3 or 7 is 32.
(d) Find the number of integers divisible by none of the numbers 3, 5, or 7:
To find the number of integers divisible by none of the numbers 3, 5, or 7, we need to subtract the number of integers divisible by at least one of these numbers from the total number of integers (300).
Number of integers divisible by at least one of 3, 5, or 7: 162
Total number of integers: 300
Number of integers divisible by none of the numbers 3, 5, or 7: 300 - 162 = 138
So, the number of integers divisible by none of the numbers 3, 5, or 7 is 138.