A six digit number is such that every alternate digit is a prime digit and the three leftmost digits forms a G.P., while last three digits (i.e., hundreds, tens and unit) form an A.P. If it is expressed as pqrstu, where p+q+r = u, q+r = t, p+r = s, r/t = 2/3 and `p ne q ne r ne s ne t ne u`, then the sum of all the digits must be:

Options

A. 25
B. 16
C. 21
D. can't be determined

Answer

To solve the problem step by step, let's break down the requirements and find the six-digit number \( pqrstu \).

### Step 1: Understand the conditions
We have the following conditions:
1. Every alternate digit is a prime digit.
2. The first three digits \( p, q, r \) form a Geometric Progression (G.P.).
3. The last three digits \( s, t, u \) form an Arithmetic Progression (A.P.).
4. The equations:
- \( p + q + r = u \)
- \( q + r = t \)
- \( p + r = s \)
- \( \frac{r}{t} = \frac{2}{3} \)
5. All digits must be distinct: \( p \neq q \neq r \neq s \neq t \neq u \).

### Step 2: Identify possible prime digits
The prime digits are: 2, 3, 5, 7. Since every alternate digit is a prime digit, we can assign \( q \) and \( r \) as prime digits.

### Step 3: Set up the G.P. condition
Let’s denote the three digits in G.P. as:
- \( p = a \)
- \( q = ar \)
- \( r = ar^2 \)

### Step 4: Set up the A.P. condition
Let’s denote the last three digits in A.P. as:
- \( s = x \)
- \( t = x + d \)
- \( u = x + 2d \)

### Step 5: Solve the ratio condition
From the condition \( \frac{r}{t} = \frac{2}{3} \):
\[ r = \frac{2}{3}t \]

### Step 6: Substitute and solve
Using \( q + r = t \):
\[ t = q + r = ar + ar^2 = ar(1 + r) \]
Substituting \( r \) into the ratio condition:
\[ ar^2 = \frac{2}{3}(ar(1 + r)) \]
This gives us a relationship between \( a \), \( r \), and \( d \).

### Step 7: Use the equations
From \( p + q + r = u \):
\[ a + ar + ar^2 = x + 2d \]
From \( p + r = s \):
\[ a + ar^2 = x \]

### Step 8: Find values for \( p, q, r, s, t, u \)
Let’s try specific values for \( p, q, r \) based on the prime digits:
1. Let \( p = 1 \), \( q = 2 \), \( r = 4 \) (since \( 1, 2, 4 \) are in G.P.).
2. Check if \( p + q + r = u \): \( 1 + 2 + 4 = 7 \) (so \( u = 7 \)).
3. Check \( q + r = t \): \( 2 + 4 = 6 \) (so \( t = 6 \)).
4. Check \( p + r = s \): \( 1 + 4 = 5 \) (so \( s = 5 \)).

### Step 9: Verify distinct digits
We have \( p = 1, q = 2, r = 4, s = 5, t = 6, u = 7 \) which are all distinct.

### Step 10: Calculate the sum of the digits
Now, we calculate the sum:
\[ 1 + 2 + 4 + 5 + 6 + 7 = 25 \]

### Conclusion
The sum of all the digits is **25**.