Understanding how many two-digit numbers are divisible by both 5 and 6 is a problem that can be approached through the concept of the Least Common Multiple (LCM). This article explains the process step-by-step and provides a comprehensive solution using various methods, including prime factorization, direct calculation, and a brute force approach in the J programming language.

Introduction to the Problem

The problem asks us to find the number of two-digit integers that are divisible by both 5 and 6. To solve this, we need to find the LCM of 5 and 6 and then identify the multiples of this LCM that fall within the two-digit range.

Prime Factorization and LCM Calculation

First, we determine the prime factorization of the numbers 5 and 6.

Prime Factorization of 5 and 6

The prime factorization of 5 is 51. The prime factorization of 6 is 21 times; 31.

The LCM of 5 and 6 is the product of the highest powers of these primes, which is

21 times; 31 times; 51 30.

The LCM of 5 and 6 is 30 because 5 and 6 have no common factors (they are coprime).

Identifying Two-Digit Multiples of 30

Now that we have the LCM, we need to find the two-digit multiples of 30. The smallest two-digit number divisible by 30 is 30, and the largest is 90. Therefore, the two-digit multiples of 30 are:

30 60 90

Counting these, we find that there are 3 such numbers.

Verification Using Brute Force in J

To further verify our solution, we can use the J programming language to perform a brute force approach.

n~./05 6/n.10 to 99

The output is:

30 60 90

This confirms that the only two-digit numbers divisible by both 5 and 6 are 30, 60, and 90, which totals to 3 numbers.

Conclusion

Therefore, the answer to the question "How many two-digit numbers are divisible by both 5 and 6?" is 3. This result is consistent whether we use prime factorization, direct calculation, or a brute force computation.