The Problem:

To find the least number that when decreased by 19 is exactly divisible by 24, 26, and 32.

Approach:

To solve this problem, we can use the concept of the least common multiple (LCM) of three numbers. The LCM of any given set of numbers is the smallest number that is divisible by all the numbers in that set.

Finding the LCM:

To find the LCM of 24, 26, and 32, we can follow these steps:

1. Prime Factorization: Begin by finding the prime factorization of each number.
- Prime factorization of 24: 2^3 * 3^1
- Prime factorization of 26: 2^1 * 13^1
- Prime factorization of 32: 2^5

2. LCM: Now, take the highest power of each prime factor from the prime factorization above.
- The highest power of 2 is 5.
- The highest power of 3 is 1.
- The highest power of 13 is 1.

3. Multiply: Multiply the highest powers of the prime factors together.
- LCM = 2^5 * 3^1 * 13^1 = 2^5 * 3 * 13 = 1248

Determining the Required Number:

To find the least number that when decreased by 19 is exactly divisible by the LCM (1248), we need to find the smallest multiple of 1248 that is 19 more than a multiple of 1248.

1. Subtract 19: Start by subtracting 19 from the LCM.
- 1248 - 19 = 1229

2. Check Divisibility: Check if 1229 is divisible by 24, 26, and 32.
- 1229 ÷ 24 = 51 remainder 5
- 1229 ÷ 26 = 47 remainder 7
- 1229 ÷ 32 = 38 remainder 5

Since 1229 is not divisible by 24, 26, and 32, we need to find the next multiple of 1248.

3. Add LCM: Add the LCM (1248) to 1229.
- 1229 + 1248 = 2477

4. Check Divisibility: Check if 2477 is divisible by 24, 26, and 32.
- 2477 ÷ 24 = 103 remainder 5
- 2477 ÷ 26 = 95 remainder 7
- 2477 ÷ 32 = 77 remainder 13

Since 2477 is not divisible by 24, 26, and 32, we continue this process until we find the least number that satisfies the given conditions.

5. Repeat Addition: Repeat the addition of the LCM (1248) to the previous result (2477).
- 2477 + 1248 = 3725

6. Check Divisibility: Check if 3725 is divisible by 24, 26, and 32.
- 3725 ÷ 24 = 155 remainder 5

The least number which when decreased by 19 is exactly divisible by 24,26,and 32is