The least number of 6-digit which when divide by 15, 25, 35, 42 and 70 respectively, leave the remainder 11, 21, 31, 38 and 66 respectively?

Options

A. 100796
B. 100256
C. 100446
D. 101046

Answer

To find the least number of 6 digits which, when divided by 15, 25, 35, 42, and 70, leaves the remainders 11, 21, 31, 38, and 66 respectively, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
- \( N \equiv 11 \mod 15 \)
- \( N \equiv 21 \mod 25 \)
- \( N \equiv 31 \mod 35 \)
- \( N \equiv 38 \mod 42 \)
- \( N \equiv 66 \mod 70 \)

### Step 2: Adjust the Remainders
To simplify the problem, we can adjust the remainders:
- Let \( N' = N - 11 \). Then we have:
- \( N' \equiv 0 \mod 15 \)
- \( N' \equiv 10 \mod 25 \)
- \( N' \equiv 20 \mod 35 \)
- \( N' \equiv 27 \mod 42 \)
- \( N' \equiv 55 \mod 70 \)

### Step 3: Find the Least Common Multiple (LCM)
Next, we need to find the least common multiple (LCM) of the divisors:
- The prime factorization of the divisors:
- \( 15 = 3 \times 5 \)
- \( 25 = 5^2 \)
- \( 35 = 5 \times 7 \)
- \( 42 = 2 \times 3 \times 7 \)
- \( 70 = 2 \times 5 \times 7 \)

The LCM is obtained by taking the highest power of each prime:
- \( LCM = 2^1 \times 3^1 \times 5^2 \times 7^1 = 2 \times 3 \times 25 \times 7 = 1050 \)

### Step 4: Find the Smallest 6-Digit Number
The smallest 6-digit number is \( 100000 \). We need to find the smallest \( N' \) such that \( N' \equiv 0 \mod 1050 \) and \( N' \) is at least \( 100000 - 11 = 99989 \).

To find the smallest multiple of 1050 that is greater than or equal to 99989:
- Calculate \( \lceil \frac{99989}{1050} \rceil \):
\[ \frac{99989}{1050} \approx 95.2 \implies \lceil 95.2 \rceil = 96 \]
- Therefore, the smallest multiple is:
\[ N' = 1050 \times 96 = 100800 \]

### Step 5: Calculate the Original Number
Now, we convert back to \( N \):
\[ N = N' + 11 = 100800 + 11 = 100811 \]

### Final Answer
The least number of 6 digits which when divided by 15, 25, 35, 42, and 70 respectively leaves the remainders 11, 21, 31, 38, and 66 is **100811**.

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