'a' divides 228 leaving a remainder 18. The biggest two digit value of 'a' is

Options

A. 70
B. 21
C. 35
D. 30

Answer

To solve the problem step by step, we need to find the largest two-digit value of 'a' that divides 228 leaving a remainder of 18.

### Step 1: Understand the division condition
When 'a' divides 228 and leaves a remainder of 18, we can express this mathematically as:
\[ 228 = k \cdot a + 18 \]
for some integer \( k \).

### Step 2: Rearranging the equation
From the equation above, we can rearrange it to find the value that is exactly divisible by 'a':
\[ 228 - 18 = k \cdot a \]
\[ 210 = k \cdot a \]
This means that \( a \) must be a divisor of 210.

### Step 3: Find the divisors of 210
To find the divisors of 210, we first need to factor it:
\[ 210 = 2 \times 3 \times 5 \times 7 \]

### Step 4: List the factors of 210
The factors of 210 can be calculated from its prime factorization:
- The factors are: 1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210.

### Step 5: Identify the two-digit factors
From the list of factors, we identify the two-digit factors:
- The two-digit factors of 210 are: 10, 14, 15, 21, 30, 35, 42, 70.

### Step 6: Find the largest two-digit factor
Among the two-digit factors listed, the largest is:
- 70.

### Conclusion
Thus, the largest two-digit value of 'a' that divides 228 leaving a remainder of 18 is:
\[ \boxed{70} \]

---