Understanding the Problem
To find the least number that, when divided by 25, 35, 45, and 55, leaves specific remainders (11, 21, 31, and 41), we can formulate the problem mathematically. The number we are looking for can be expressed in the form:
- x ≡ 11 (mod 25)
- x ≡ 21 (mod 35)
- x ≡ 31 (mod 45)
- x ≡ 41 (mod 55)
Step 1: Adjusting the Equations
We can adjust each equation to eliminate the remainders:
- x - 11 = 25k
- x - 21 = 35m
- x - 31 = 45n
- x - 41 = 55p
Where k, m, n, and p are integers.
This means we can express x in terms of these equations:
- x = 25k + 11
- x = 35m + 21
- x = 45n + 31
- x = 55p + 41
Step 2: Finding the LCM
Next, to solve these congruences, we find the least common multiple (LCM) of the divisors (25, 35, 45, 55):
- LCM(25, 35, 45, 55) = 3465
Step 3: Setting Up the System of Equations
Now, we need to find a common solution based on the adjustments made earlier.
Step 4: Solving the System
Start testing values of x that fit one of the equations and check if they satisfy the other conditions:
- x = LCM + remainder = 3465 + 11 = 3476
Check with other equations:
- 3476 mod 35 = 21 (correct)
- 3476 mod 45 = 31 (correct)
- 3476 mod 55 = 41 (correct)
Conclusion
Thus, the least number that meets all the conditions is:
- 3476