Let x be the least number which when divided by 10, 12, 14 and 16 leaves the remainders 2, 4, 6 and 8, respectively, but x is divisible by 17. Whenx is divided by 52, the quotient is:

Options

A. 96
B. 92
C. 95
D. 99

Answer

To solve the problem step by step, we need to find the least number \( x \) that satisfies the given conditions.

### Step 1: Understand the Remainders
The problem states that when \( x \) is divided by 10, 12, 14, and 16, it leaves remainders of 2, 4, 6, and 8, respectively. We can express these conditions mathematically:
- \( x \equiv 2 \mod 10 \)
- \( x \equiv 4 \mod 12 \)
- \( x \equiv 6 \mod 14 \)
- \( x \equiv 8 \mod 16 \)

### Step 2: Rewrite the Congruences
We can rewrite these congruences by subtracting the remainders:
- \( x - 2 \equiv 0 \mod 10 \) → \( x - 2 = 10k_1 \)
- \( x - 4 \equiv 0 \mod 12 \) → \( x - 4 = 12k_2 \)
- \( x - 6 \equiv 0 \mod 14 \) → \( x - 6 = 14k_3 \)
- \( x - 8 \equiv 0 \mod 16 \) → \( x - 8 = 16k_4 \)

This means that \( x \) can be expressed as:
- \( x = 10k_1 + 2 \)
- \( x = 12k_2 + 4 \)
- \( x = 14k_3 + 6 \)
- \( x = 16k_4 + 8 \)

### Step 3: Find the LCM
Next, we find the least common multiple (LCM) of the divisors (10, 12, 14, and 16) to find a common structure for \( x \):
- The prime factorization of the numbers:
- \( 10 = 2 \times 5 \)
- \( 12 = 2^2 \times 3 \)
- \( 14 = 2 \times 7 \)
- \( 16 = 2^4 \)

The LCM is obtained by taking the highest power of each prime:
- LCM = \( 2^4 \times 3^1 \times 5^1 \times 7^1 = 1680 \)

### Step 4: General Form of \( x \)
Since \( x \) must leave a remainder of 8 when divided by 16, we can express \( x \) in the form:
\[ x = 1680k - 8 \]

### Step 5: Ensure Divisibility by 17
We also need \( x \) to be divisible by 17:
\[ 1680k - 8 \equiv 0 \mod 17 \]
Calculating \( 1680 \mod 17 \):
\[ 1680 \div 17 \approx 98.8235 \Rightarrow 98 \times 17 = 1666 \Rightarrow 1680 - 1666 = 14 \]
Thus:
\[ 14k - 8 \equiv 0 \mod 17 \]
This simplifies to:
\[ 14k \equiv 8 \mod 17 \]
To solve for \( k \), we can find the multiplicative inverse of 14 modulo 17. The inverse is 10 since:
\[ 14 \times 10 \equiv 1 \mod 17 \]
Multiplying both sides of \( 14k \equiv 8 \) by 10:
\[ k \equiv 80 \mod 17 \Rightarrow k \equiv 12 \mod 17 \]

### Step 6: Find the Least Value of \( k \)
The smallest positive integer \( k \) that satisfies this is \( k = 12 \).

### Step 7: Calculate \( x \)
Substituting \( k = 12 \) back into the equation for \( x \):
\[ x = 1680 \times 12 - 8 = 20160 - 8 = 20152 \]

### Step 8: Divide \( x \) by 52
Now we divide \( x \) by 52 to find the quotient:
\[ 20152 \div 52 = 387 \]
Thus, the quotient is 387.

### Final Answer
The quotient when \( x \) is divided by 52 is **387**.