Question

How many integers between 999 and 9999 either begin or end with 3 ?​

Answer 1

Step-by-step explanation:

The integers which are between 999 and 9999 and divisible by 2 are the even 4-digit numbers. There are (9998/2 - 998/2) or (10000/2 - 1000/2) = 5000 - 500

= 4500 such numbers.

The integers which are between 999 and 9999 and divisible by 5 are the 4-digit numbers divisible by 5. There are (9995/5 - 995/5) or (10000/5 - 1000/5) = 2000 - 200

= 1800 such numbers.

The integers which are between 999 and 9999 and divisible by both 2 and 5 are the 4-digit numbers divisible by 2x5 = 10. There are (9990/10 - 990/10) or (10000/10 - 1000/10) = 1000 - 100

= 900 such numbers.

So, using the Principle of Inclusion and Exclusion, we can say that the integers which are between 999 and 9999 and divisible by either 2 and 5 are the 4-digit numbers divisible by 2 plus the 4-digit numbers divisible by 5 minus the 4-digit numbers divisible by both 2 and 5 (i.e divisible by 10). This is equal to 4500 + 1800 - 900

= 5400.

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