How many positive integers between 1000 and 9999 inclusive?

(a) are divisible by nine?

(b) are even?

(c) have distinct digits?

(d) are not divisible by 3?

(e) are divisible by either 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?

(a) 1000

(b) 4500

(c) 4536

(d) 3000

(e) 2829

(f) 6171

(g) 1543

(h) 257

Step-by-step explanation:

Let X be positive integers between 1000 and 9999 which contains 9000 integers.

(a) The integers divided by the number of elements 9 is:

Use the quotient rule = absolute number of integers (X) / number of elements

i = |9000|/9 = 1000

(b) Similar we use the quotient rule but we use an even number since even numbers are divided by 2 we can use 2.

i = |9000|/2 = 4500

(c) There are 10 digits

The first digit cannot be be zero so you can divide is 9 ways

The second digit can be zero but not the same as the first digit and therefore 9 ways

The third digit 8 ways because the second digit cannot be the same as the first and second.

The fourth digit 7 ways for the same reason as the third digit.

Using the product rule = i = 9*9*8*7 = 4536 have distinct digits

(d) Use the quotient rule = i = 9000/3 = 3000

(e) Use the quotient rule = i5 = 9000/5 = 1800

Use the quotient rule = i7 = 9000/7 = 1286

the integers divisible by 5 and 7 is 5*7 = 35

i35 = 9000/35 = 257

Number of integers divisible by 5 or 7 can be determined by the subtraction rule therefore:

i = i5+i7-i35 = 1800+1286-257 = 2829

(f) The integers divisible not divided 5 or 7 is the same as the integers not divisible by 3 or 4:

i (not 5 or 7) = |X| - i (5 or 7) = 9000-2829 = 6171

(g) Integers divisible by 11 but not 7 are the same as integers divisible by 11 but not divisible by 7 and 11.

i (5 not by 7) = i5-i35 = 1800 - 257 = 1543

(h) i (5 and 7) = i35 = 257