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Question

How many numbers between 1 and 100 (inclusive) are divisible by 3 or 4?


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Solution

Step-1: Find the number between 1 to 100 divisible by 3.

The first number is 3 and the last number is 99.

Let the total number be n and the common difference is 3.

Recall the nth term of the arithmetic formula:

an=a+(n-1)d

Step-2: Find the number of terms.

Substitute a=3,d=3,an=99:

99=3+(n-1)399-3=(n-1)396=(n-1)3n-1=963n=32+1n=33

Thus, the number of terms divisible by 3 is ndivisibleby3=33.

Step 2: Find the number between 1 to 100 divisible by 4.

Substitute a=4,d=4,an=100 in an=a+(n-1)d:

99=3+(n-1)3100-4=(n-1)496=(n-1)4n-1=964n=24+1n=25

Thus, the number of terms divisible by 4is ndivisibleby4=25.

Step 2: Find the number between 1 to 100 divisible by 3and4.

The LCM of 3and4is12.

Substitute a=12,d=12,an=96 in an=a+(n-1)d:

99=3+(n-1)396-12=(n-1)1284=(n-1)12n-1=8412n=7+1n=8

Similarly, the number of terms divisible by 12 is ndivisibleby3and4=8.

The total number of terms between 1 to 100 are divisible by 3 or 4:

ndivisibleby3or4=ndivisibleby3+ndivisibleby3+ndivisibleby3and4ndivisibleby3or4=33+25-8ndivisibleby3or4=50

Hence, the total number between 1 to 100 divisible by 3or4 are 50.


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