Divisibility by 3: How Many Numbers from 10 to 50 are Exactly Divisible?

Divisibility by 3 is a fundamental concept in number theory. Understanding how many numbers within a specific range are exactly divisible by 3 can provide insights into various mathematical problems. In this article, we will explore how many numbers from 10 to 50 are exactly divisible by 3, and we'll break down the steps to find the solution.

Introduction to Divisibility by 3

Divisibility by 3 is a property of numbers that can be determined by a simple rule. A number is exactly divisible by 3 if the sum of its digits is also divisible by 3. For example, 123 is divisible by 3 because 1 2 3 6, and 6 is divisible by 3.

Identifying Numbers Exactly Divisible by 3 from 10 to 50

Let's start by simplifying the range from 10 to 50 to make the task more manageable. We know that any number divisible by 3 will end in one of the following digits: 0, 3, 6, or 9.

10 (not divisible by 3) 11 (not divisible by 3) 12 (divisible by 3) 13 (not divisible by 3) 14 (not divisible by 3) 15 (divisible by 3)

Following this pattern, we can list the numbers from 10 to 50 that are exactly divisible by 3:

12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48

Counting these numbers, we find there are 13 numbers that are exactly divisible by 3 in this range.

Calculation Method

A more systematic approach to solving this problem involves using the division method. We can use the formula to find the number of integers exactly divisible by a given number within a specific range. For divisibility by 3, we can use two main steps:

Divide the lower limit of the range by 3 and round up to the nearest integer. Divide the upper limit of the range by 3 and round down to the nearest integer. Subtract the result of the first division from the second and add 1 to get the count of numbers exactly divisible by 3.

For the range from 10 to 50:

10 div; 3 3 remainder 1 (rounded up to 4)

50 div; 3 16 remainder 2 (rounded down to 16)

The number of integers from 10 to 50 exactly divisible by 3 is 16 - 4 1 13.

Conclusion

From this exploration, we've discovered that there are 13 numbers from 10 to 50 that are exactly divisible by 3. Whether you're working on a math problem or just curious about divisibility rules, understanding how many numbers fit into specific ranges can be quite useful.

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