How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

Solution:

Given, the integers from 0 to 50,

We have to find the integers divided by 3 that leaves a remainder 1.

The integers from 0 to 50 when divided by 3 leaves remainder 0 are 3,6,9,12,15,18,21,24,27,30,33,36,39,42,45,48.

The numbers are in arithmetic progression with common difference 3.

Now, the integers from 0 to 50 when divided by 3 leaves remainder 1 are 4,7,10,...49.

We can use arithmetic progression to find the number of terms,

Here, a = 4 and d = 3

Tₙ = a + (n-1)d

49 = 4 + (n-1)3

49 = 4 + 3n - 3

49 = 3n + 1

3n = 49 - 1

3n = 48

n = 48/3

n = 16

Therefore, there are 16 integers from 0 to 50.

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How many integers from 0 to 50, inclusive, have a remainder of 1 when divided by 3 ?

Summary:

The number of integers from 0 to 50, inclusive, that leaves a remainder of 1 when divided by 3 is 16.