To find the least number that leaves the same remainder when divided by 5, 7, 9, and 12, we can follow a systematic approach involving the Least Common Multiple (LCM).
Understanding the Problem
We need to find a number \( N \) such that:
- \( N \mod 5 = r \)
- \( N \mod 7 = r \)
- \( N \mod 9 = r \)
- \( N \mod 12 = r \)
Where \( r \) is the common remainder.
Transforming the Problem
This can be rewritten as:
- \( N - r \) is divisible by 5, 7, 9, and 12.
Let \( M = N - r \). Thus, \( M \) must be a common multiple of these numbers.
Finding the LCM
We calculate the LCM of 5, 7, 9, and 12:
- Prime Factorization:
- \( 5 = 5^1 \)
- \( 7 = 7^1 \)
- \( 9 = 3^2 \)
- \( 12 = 2^2 \times 3^1 \)
- Taking the Highest Powers:
- \( 2^2, 3^2, 5^1, 7^1 \)
- Calculating the LCM:
\[
LCM = 2^2 \times 3^2 \times 5^1 \times 7^1 = 4 \times 9 \times 5 \times 7
\]
Calculating step-by-step:
- \( 4 \times 9 = 36 \)
- \( 36 \times 5 = 180 \)
- \( 180 \times 7 = 1260 \)
So, \( LCM(5, 7, 9, 12) = 1260 \).
Finding the Least Number
The least number \( N \) which leaves the same remainder \( r \) can be expressed as:
\[
N = 1260 + r
\]
To find the least such number, we can set \( r = 0 \):
\[
N = 1260
\]
Thus, the least number that leaves the same remainder when divided by 5, 7, 9, and 12 is:
Final Answer:
1260