Find `150%` of x, If x is the least number which when divided by 6, 7, 8, 9 and 12 leaves remainders 2, 3, 4, 5 and 8 respectively
A. 750
B.500
C.1000
D.1260

Options

A. B
B. D
C. A
D. C

Answer

To solve the problem step by step, we need to find the least number \( x \) that satisfies the given conditions and then calculate \( 150\% \) of that number.

### Step 1: Understand the problem
We need to find a number \( x \) such that:
- When \( x \) is divided by 6, the remainder is 2.
- When \( x \) is divided by 7, the remainder is 3.
- When \( x \) is divided by 8, the remainder is 4.
- When \( x \) is divided by 9, the remainder is 5.
- When \( x \) is divided by 12, the remainder is 8.

### Step 2: Set up the equations
From the conditions, we can express \( x \) in terms of the divisors and remainders:
- \( x \equiv 2 \mod 6 \)
- \( x \equiv 3 \mod 7 \)
- \( x \equiv 4 \mod 8 \)
- \( x \equiv 5 \mod 9 \)
- \( x \equiv 8 \mod 12 \)

### Step 3: Rewrite the equations
To simplify, we can rewrite these congruences:
- \( x = 6k + 2 \) for some integer \( k \)
- \( x = 7m + 3 \) for some integer \( m \)
- \( x = 8n + 4 \) for some integer \( n \)
- \( x = 9p + 5 \) for some integer \( p \)
- \( x = 12q + 8 \) for some integer \( q \)

### Step 4: Adjust the equations
To find a common solution, we can adjust each equation to have the same form:
- \( x - 2 \equiv 0 \mod 6 \)
- \( x - 3 \equiv 0 \mod 7 \)
- \( x - 4 \equiv 0 \mod 8 \)
- \( x - 5 \equiv 0 \mod 9 \)
- \( x - 8 \equiv 0 \mod 12 \)

This means we can define a new variable \( y \):
- \( y = x - 2 \)
Then we have:
- \( y \equiv 0 \mod 6 \)
- \( y \equiv 1 \mod 7 \)
- \( y \equiv 2 \mod 8 \)
- \( y \equiv 3 \mod 9 \)
- \( y \equiv 6 \mod 12 \)

### Step 5: Find the least common multiple (LCM)
Now we need to find the least common multiple of the divisors (6, 7, 8, 9, and 12).

Calculating the LCM:
- The prime factorization is:
- \( 6 = 2 \times 3 \)
- \( 7 = 7 \)
- \( 8 = 2^3 \)
- \( 9 = 3^2 \)
- \( 12 = 2^2 \times 3 \)

The LCM is obtained by taking the highest power of each prime:
- \( LCM = 2^3 \times 3^2 \times 7 = 8 \times 9 \times 7 = 504 \)

### Step 6: Calculate the least number \( x \)
Now, since \( y \) must be of the form \( 504k \) for some integer \( k \):
- The least value of \( x \) can be found by substituting \( k = 1 \):
- \( y = 504 \)
- Therefore, \( x = y + 2 = 504 + 2 = 506 \).

### Step 7: Calculate \( 150\% \) of \( x \)
Now, we need to calculate \( 150\% \) of \( x \):
\[ 150\% \text{ of } 506 = \frac{150}{100} \times 506 = 1.5 \times 506 = 759 \]

### Final Answer
Thus, the answer is \( 759 \).