To find the least number that satisfies the given condition, you need to:
1. Determine the least common multiple (LCM) of the numbers 36, 40, and 64 because the number we are looking for must be divisible by each of these numbers once it is increased by 3.
2. Find a multiple of the LCM that, when reduced by 3, yields a positive number.
Firstly, let's find the LCM of 36, 40, and 64 manually:
- The prime factorization of 36 is \(2^2 \times 3^2\).
- The prime factorization of 40 is \(2^3 \times 5\).
- The prime factorization of 64 is \(2^6\).
The LCM is calculated by taking the highest power of each prime number present in the factorizations:
- For the prime number 2, the highest power is \(2^6\) (from 64).
- For the prime number 3, the highest power is \(3^2\) (from 36).
- For the prime number 5, the highest power is \(5\) (from 40).
Therefore, the LCM of 36, 40, and 64 is \(2^6 \times 3^2 \times 5 = 64 \times 9 \times 5\).
Now, calculate the product:
\(64 \times 9 \times 5 = 576 \times 5 = 2880\).
So, the LCM of 36, 40, and 64 is 2880.
We need to find the smallest number that is one less than a multiple of 2880, because when you add 3 to it, it will become a multiple of 2880. Mathematics doesn't require us to check all multiples of 2880 to find the result. We know that adding 3 to a multiple of 2880 will yield another multiple of 2880, so what we actually need to do is find the least non-negative number that is 3 less than a multiple of 2880.
Since our LCM is 2880, any multiple of 2880 minus 3 will be negative for the first factor (which is just -3). All other factors of 2880, when reduced by 3, will result in a valid positive number.
Thus, the least number we're looking for is the first positive multiple of 2880, less 3. We can express this as:
\(1 \times 2880 - 3 = 2880 - 3 = 2877\).
Therefore, the least number we are looking for is 2877, which when increased by 3, is divisible by 36, 40, and 64.
