To find the least number that leaves a remainder of 5 when divided by 16, 34, and 40, follow these steps:
Understanding the Problem
We are looking for a number, let’s call it \( x \), such that:
- \( x \mod 16 = 5 \)
- \( x \mod 34 = 5 \)
- \( x \mod 40 = 5 \)
This means that when \( x \) is divided by each of these numbers, it leaves a remainder of 5.
Formulating the Equations
We can rewrite the conditions as:
- \( x = 16k + 5 \)
- \( x = 34m + 5 \)
- \( x = 40n + 5 \)
Where \( k, m, n \) are integers.
Finding the Common Multiple
To solve for \( x \), we can subtract 5 from each equation:
- \( x - 5 \) must be divisible by 16, 34, and 40.
Let’s denote \( y = x - 5 \). Thus, we need to find the least common multiple (LCM) of 16, 34, and 40.
Calculating the LCM
1. Prime factorization:
- \( 16 = 2^4 \)
- \( 34 = 2^1 \times 17^1 \)
- \( 40 = 2^3 \times 5^1 \)
2. Taking the highest powers:
- LCM = \( 2^4 \times 5^1 \times 17^1 \)
Calculating this gives:
- \( LCM = 16 \times 5 \times 17 = 1360 \)
Final Calculation
Now add back the 5:
- \( x = y + 5 = 1360 + 5 = 1365 \)
Conclusion
The least number which, when divided by 16, 34, and 40, leaves a remainder of 5 is:
- \( 1365 \)