How many positive integers between 1000 and 9999 inclusive...
a) are divisible by 9?
b) are even?
c) have distinct digits?
d) are not divisible by 3?
e) are divisible by 5 or 7?
f) are not divisible by either 5 or 7?
g) are divisible by 5 but not by 7?
h) are divisible by 5 and 7?
First, we need to know that there are 9999-1000+1=9000 integers in our question.
a) Since 9, 18,... are divisible by 9, we can see that exactly every 9th one is divisible by 9, there are 9000/9=1000 integers which are divisible by 9.
b) Since there are half even and half odd integers between 1000 and 9999 inclusive, there are 9000/2=4500 integers which are even.
c) Since we have 9 choices for first digit (from 1 to 9), 10-1=9 choices for second (from 0 to 9 but exclude first digit), 10-2 choices for third (from 0 to 9 ...
In this solution we count positive in integers between 1000 and 9999 based on a number of factors.