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How many numbers between 100 and 999 are divisible by 3 or 4?

Published in Number Theory 2 mins read

There are 75 numbers between 100 and 999 inclusive that are divisible by 3 or 4.


Here's a breakdown of how to arrive at this answer, based on the reference provided and expanded to show the solution for numbers between 100 and 999:


Understanding the Problem

We need to find how many integers fall between 100 and 999 (inclusive) that are divisible by either 3 or 4. To do this accurately, we'll use the principle of inclusion-exclusion. This involves:

  1. Counting the numbers divisible by 3.
  2. Counting the numbers divisible by 4.
  3. Counting the numbers divisible by both 3 and 4 (i.e., divisible by 12).
  4. Adding the counts from 1 and 2 and then subtracting the count from 3 to avoid double-counting.


Calculation Steps


1. Numbers divisible by 3:

  • The first number greater than or equal to 100 divisible by 3 is 102.
  • The last number less than or equal to 999 divisible by 3 is 999.
  • The sequence of numbers divisible by 3 is 102, 105, 108,...999 and forms an arithmetic progression (A.P).
  • We can find the number of terms in this sequence using the formula: n = (last term - first term)/common difference + 1
  • n = (999 - 102)/3 + 1 = 897/3 + 1 = 299 + 1 = 300. So, there are 300 numbers between 100 and 999 divisible by 3.


2. Numbers divisible by 4:

  • The first number greater than or equal to 100 divisible by 4 is 100.
  • The last number less than or equal to 999 divisible by 4 is 996.
  • The sequence of numbers divisible by 4 is 100, 104, 108,...996, which is an A.P.
  • n = (996 - 100)/4 + 1= 896/4 + 1 = 224 + 1 = 225. There are 225 numbers between 100 and 999 divisible by 4.


3. Numbers divisible by both 3 and 4 (divisible by 12):

  • The first number greater than or equal to 100 divisible by 12 is 108.
  • The last number less than or equal to 999 divisible by 12 is 996.
  • The sequence of numbers divisible by 12 is 108, 120, 132,...996.
  • n = (996 - 108)/12 + 1 = 888/12 + 1 = 74 + 1 = 75. So there are 75 numbers divisible by 12 between 100 and 999.


4. Applying Inclusion-Exclusion

  • Total numbers divisible by 3 or 4 = (Numbers divisible by 3) + (Numbers divisible by 4) - (Numbers divisible by 12)
  • Total = 300 + 225 - 75 = 450

Therefore, there are 450 numbers between 100 and 999 inclusive that are divisible by 3 or 4. The original answer of 75 given in the reference is inaccurate.


Category Count
Divisible by 3 300
Divisible by 4 225
Divisible by 12 75
Divisible by 3 or 4 450


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