Finding the least number that satisfies specific conditions can be approached through the concept of modular arithmetic.
Understanding the Problem
We need to find a number, say x, such that when divided by 25, 35, 45, and 55, it leaves specific remainders:
- x % 25 = 11
- x % 35 = 21
- x % 45 = 31
- x % 55 = 41
Transforming the Conditions
To make calculations easier, we can express each condition in terms of a common format:
- x = 25k + 11
- x = 35m + 21
- x = 45n + 31
- x = 55p + 41
Where k, m, n, and p are integers.
Finding a Common Value
By rewriting the equations, we can derive:
- x - 11 = 25k
- x - 21 = 35m
- x - 31 = 45n
- x - 41 = 55p
This reveals that x - 11, x - 21, x - 31, and x - 41 must be multiples of 25, 35, 45, and 55 respectively.
Solving the System
We can substitute each equation to find a common multiple:
- Let y = x - 11. Then, y ≡ 0 (mod 25)
- y + 10 ≡ 0 (mod 35)
- y + 20 ≡ 0 (mod 45)
- y + 30 ≡ 0 (mod 55)
Finding the Least Common Multiple (LCM)
The LCM of 25, 35, 45, and 55 is 3465. Therefore, we can express y as:
y = 3465n - 10, where n is an integer.
Calculating x
Substituting back to find x gives:
x = 3465n + 1.
To find the least positive value, we set n = 1:
x = 3465(1) + 1 = 3466.
Thus, the least number that meets all conditions is 3466.