Which number lying between 800 and 900 is such that when it is divided by either of 34 and 51, the remainder left is 19 ?

Options

A. 887
B. 816
C. 835
D. 869

Answer

To find the number between 800 and 900 that leaves a remainder of 19 when divided by both 34 and 51, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
- \( N \equiv 19 \mod 34 \)
- \( N \equiv 19 \mod 51 \)

### Step 2: Set Up the Equations
From the above congruences, we can express \( N \) in terms of the least common multiple (LCM) of 34 and 51:
- \( N = k \cdot \text{LCM}(34, 51) + 19 \)

### Step 3: Calculate the LCM of 34 and 51
To find the LCM, we can use the prime factorization method:
- \( 34 = 2 \times 17 \)
- \( 51 = 3 \times 17 \)

The LCM is found by taking the highest power of each prime:
- LCM(34, 51) = \( 2^1 \times 3^1 \times 17^1 = 102 \)

### Step 4: Substitute the LCM into the Equation
Now, we substitute the LCM back into our equation for \( N \):
- \( N = k \cdot 102 + 19 \)

### Step 5: Find Values of \( k \) for \( N \) between 800 and 900
We need to find integer values of \( k \) such that:
- \( 800 < k \cdot 102 + 19 < 900 \)

Subtracting 19 from all parts:
- \( 781 < k \cdot 102 < 881 \)

Now, divide by 102:
- \( \frac{781}{102} < k < \frac{881}{102} \)

Calculating the bounds:
- \( \frac{781}{102} \approx 7.66 \)
- \( \frac{881}{102} \approx 8.63 \)

Since \( k \) must be an integer, the only possible value for \( k \) is 8.

### Step 6: Calculate \( N \) Using \( k = 8 \)
Now, substitute \( k = 8 \) back into the equation for \( N \):
- \( N = 8 \cdot 102 + 19 \)
- \( N = 816 + 19 = 835 \)

### Step 7: Conclusion
The number we found is \( N = 835 \), which lies between 800 and 900.

### Final Answer
The required number is **835**.

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