Each of two arithmetic progressions 2, 4, 6, ... and 3, 6, 9, ... are taken upto 200 terms. How many terms are common in these two progressions ?

Explanation

To find the number of common terms between the two arithmetic progressions (APs) given, we will follow these steps:

### Step 1: Identify the first arithmetic progression (AP)
The first AP is given as:
\[ 2, 4, 6, \ldots \]
This AP has:
- First term \( a_1 = 2 \)
- Common difference \( d_1 = 4 - 2 = 2 \)

### Step 2: Find the last term of the first AP
To find the last term of the first AP when taken up to 200 terms, we use the formula for the \( n \)-th term of an AP:
\[ a_n = a + (n - 1) \cdot d \]
For the 200th term:
\[ a_{200} = 2 + (200 - 1) \cdot 2 \]
\[ a_{200} = 2 + 199 \cdot 2 \]
\[ a_{200} = 2 + 398 = 400 \]

### Step 3: Identify the second arithmetic progression (AP)
The second AP is given as:
\[ 3, 6, 9, \ldots \]
This AP has:
- First term \( b_1 = 3 \)
- Common difference \( d_2 = 6 - 3 = 3 \)

### Step 4: Find the last term of the second AP
To find the last term of the second AP when taken up to 200 terms:
\[ b_{200} = 3 + (200 - 1) \cdot 3 \]
\[ b_{200} = 3 + 199 \cdot 3 \]
\[ b_{200} = 3 + 597 = 600 \]

### Step 5: Determine the common terms
Now we need to find the common terms between the two APs. The first AP has terms of the form:
\[ 2, 4, 6, 8, 10, \ldots \]
The second AP has terms of the form:
\[ 3, 6, 9, 12, 15, \ldots \]

The common terms will be the multiples of 6, since:
- The first AP has every second number starting from 2.
- The second AP has every third number starting from 3.

### Step 6: Identify the common AP
The common terms can be expressed as:
\[ 6, 12, 18, \ldots \]
This is another AP where:
- First term \( c_1 = 6 \)
- Common difference \( d_c = 6 \)

### Step 7: Find the last common term less than or equal to 400
To find the last term of this common AP that is less than or equal to 400, we can express the \( n \)-th term of this common AP:
\[ c_n = 6n \]
We need to find the largest \( n \) such that:
\[ 6n \leq 400 \]
Dividing both sides by 6:
\[ n \leq \frac{400}{6} \]
Calculating this gives:
\[ n \leq 66.67 \]
Thus, the largest integer \( n \) is 66.

### Step 8: Conclusion
Therefore, the number of common terms between the two arithmetic progressions is:
\[ \boxed{66} \]