To find the least number consisting of five digits which, when divided by 6, 7, 8, 9, and 10, leaves remainders of 4, 5, 6, 7, and 8 respectively, we can follow these steps:
### Step 1: Understand the Problem
We need to find a five-digit number \( N \) such that:
- \( N \equiv 4 \mod 6 \)
- \( N \equiv 5 \mod 7 \)
- \( N \equiv 6 \mod 8 \)
- \( N \equiv 7 \mod 9 \)
- \( N \equiv 8 \mod 10 \)
### Step 2: Rewrite the Congruences
We can rewrite the congruences in a more manageable form:
- \( N - 4 \equiv 0 \mod 6 \)
- \( N - 5 \equiv 0 \mod 7 \)
- \( N - 6 \equiv 0 \mod 8 \)
- \( N - 7 \equiv 0 \mod 9 \)
- \( N - 8 \equiv 0 \mod 10 \)
This means that \( N - 4 \) is a common multiple of 6, 7, 8, 9, and 10.
### Step 3: Find the Least Common Multiple (LCM)
To solve for \( N - 4 \), we need to find the least common multiple (LCM) of the divisors:
- The prime factorization of the numbers is:
- \( 6 = 2 \times 3 \)
- \( 7 = 7 \)
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 10 = 2 \times 5 \)
The LCM will take the highest power of each prime:
- \( 2^3 \) from 8
- \( 3^2 \) from 9
- \( 5^1 \) from 10
- \( 7^1 \) from 7
Calculating the LCM:
\[
\text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 8 \times 9 \times 5 \times 7 = 2520
\]
### Step 4: Set Up the Equation
Now we know that:
\[
N - 4 = k \times 2520
\]
for some integer \( k \). Therefore:
\[
N = k \times 2520 + 4
\]
### Step 5: Find the Smallest Five-Digit Number
The smallest five-digit number is 10,000. We need to find the smallest \( k \) such that:
\[
k \times 2520 + 4 \geq 10000
\]
Rearranging gives:
\[
k \times 2520 \geq 9996
\]
\[
k \geq \frac{9996}{2520} \approx 3.968
\]
Thus, the smallest integer \( k \) is 4.
### Step 6: Calculate \( N \)
Now, substituting \( k = 4 \):
\[
N = 4 \times 2520 + 4 = 10080 + 4 = 10084
\]
### Step 7: Verify the Conditions
We need to check if \( N = 10084 \) satisfies the original conditions:
- \( 10084 \div 6 = 1680 \) remainder \( 4 \)
- \( 10084 \div 7 = 1439 \) remainder \( 5 \)
- \( 10084 \div 8 = 1260 \) remainder \( 6 \)
- \( 10084 \div 9 = 1120 \) remainder \( 7 \)
- \( 10084 \div 10 = 1008 \) remainder \( 8 \)
All conditions are satisfied.
### Final Answer
Thus, the least number consisting of five digits that meets the criteria is **10084**.
