Answer to Question #140317 in Discrete Mathematics for Promise Omiponle

Question

Find how many positive integers with exactly four decimal digits, that is, positive integers between 1000 and 9999 inclusive, have the following properties:
(a) have distinct digits.
(b) are divisible by 5 and by 7.
(c) are even.
(d) are not divisible by either 5 or 7.

Answer 1

Answer a)

First write down all the 4 digit numbers in excel sheet and then we can select all the numbers which do not have same digits as the numbers highlighted in yellow.

For example 1023 which has digits 1, 0, 2 & 3 where none of the digits are same.

Hence there are total 4536 integers between 1000 & 9999 which have distinct digits.

Answer b) Since we have to find the total number of positive integers divisible by 5 & by 7 that means they should be divisible by 5*7 = 35. So after the number 1000, 1015 is the first number which is divisible by 35. 1015/35 = 29

Similarly, 9975 is the last number before 9999 which is divisible by 35 i.e 9975/35 = 285

Hence all the numbers from 29 to 285 when multiplied by 35 will give you a positive integer between 1000 & 9999.

So there are total 257 integers (285 -29 + 1= 257) which are divisible by 35 (i.e divisible by both 5 & by 7) from 1000 & 9999

Answer c) There are total 9000 numbers between 1000 & 9999 but every second number is even. This means out of total 9000 numbers half the numbers are even.

Hence 9000/2 = 4500.

Total there are 4500 integers between 1000 & 9999 which are even

Answer d) Total number of integers that are divisible by 5 are 1800 and by 7 are 1285.

But there are 247 integers which are divisible by both 5 & by 7 hence the total number of integers that are not divisible by either 5 or 7 = 9000 - 1800 - 1285 + 247 = 6162