Key Concepts and Formulas

• Arithmetic Progression (A.P.): A sequence of numbers such that the difference between the consecutive terms is constant. The general form is $a, a+d, a+2d, \ldots$, where $a$ is the first term and $d$ is the common difference. The $n$-th term is given by $a_n = a + (n-1)d$.
• Common Terms of Two A.P.s: The terms common to two A.P.s themselves form an A.P.
  • The first term of this new A.P. is the smallest term present in both original A.P.s.
  • The common difference of this new A.P. is the Least Common Multiple (LCM) of the common differences of the two original A.P.s.
• Finding the Number of Terms: If the new A.P. has first term $A$ and common difference $D$, and its terms must be less than or equal to a maximum value $M$, the number of terms $k$ is found by solving $A + (k-1)D \le M$ for the largest integer $k$.

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Step-by-Step Solution

Step 1: Analyze the First Arithmetic Progression (A.P.) We are given the first A.P. as $3, 7, 11, \ldots, 407$.

• The first term ($a_1$) is $3$.
• The common difference ($d_1$) is $7 - 3 = 4$.
• The last term is $407$. The general term of this A.P. is $T_n = a_1 + (n-1)d_1 = 3 + (n-1)4$.

Step 2: Analyze the Second Arithmetic Progression (A.P.) We are given the second A.P. as $2, 9, 16, \ldots, 709$.

• The first term ($a_2$) is $2$.
• The common difference ($d_2$) is $9 - 2 = 7$.
• The last term is $709$. The general term of this A.P. is $T_m = a_2 + (m-1)d_2 = 2 + (m-1)7$.

Step 3: Find the First Common Term To find the terms common to both A.P.s, we can look for a number that can be expressed in the form $3 + (n-1)4$ and also in the form $2 + (m-1)7$ for some positive integers $n$ and $m$. Let's list the initial terms of both sequences:

• A.P. 1: $3, 7, 11, 15, 19, 23, 27, 31, \ldots$
• A.P. 2: $2, 9, 16, 23, 30, 37, \ldots$ By comparing the terms, we can see that the first common term is $23$. We can verify this: For A.P. 1: $23 = 3 + (n-1)4 \Rightarrow 20 = (n-1)4 \Rightarrow n-1 = 5 \Rightarrow n=6$. For A.P. 2: $23 = 2 + (m-1)7 \Rightarrow 21 = (m-1)7 \Rightarrow m-1 = 3 \Rightarrow m=4$. Since we found integer values for $n$ and $m$, $23$ is indeed a common term.

Step 4: Find the Common Difference of the New A.P. of Common Terms The common terms of two A.P.s form a new A.P. The common difference of this new A.P. is the Least Common Multiple (LCM) of the common differences of the original A.P.s.

• $d_1 = 4$
• $d_2 = 7$ The LCM of $4$ and $7$ is $\text{LCM}(4, 7)$. Since $4$ and $7$ are coprime (their greatest common divisor is $1$), their LCM is their product: $D = \text{LCM}(4, 7) = 4 \times 7 = 28$. So, the common difference of the new A.P. is $28$.

Step 5: Determine the Upper Bound for the Common Terms A term common to both A.P.s must exist within the range of both original A.P.s. Therefore, any common term cannot exceed the smaller of the last terms of the two original A.P.s.

• Last term of A.P. 1: $407$
• Last term of A.P. 2: $709$ The upper bound for the common terms is $\min(407, 709) = 407$.

Step 6: Find the Number of Common Terms The new A.P. of common terms has:

• First term ($A$) = $23$
• Common difference ($D$) = $28$ Let $k$ be the number of terms in this new A.P. The $k$-th term is given by $A_k = A + (k-1)D$. We need to find the number of terms $k$ such that $A_k \le 407$. $23 + (k-1)28 \le 407$ Subtract $23$ from both sides: $(k-1)28 \le 407 - 23$ $(k-1)28 \le 384$ Divide by $28$: $k-1 \le \frac{384}{28}$ Simplify the fraction: $k-1 \le \frac{96}{7}$ Calculate the decimal value: $k-1 \le 13.714\ldots$ Add $1$ to both sides: $k \le 14.714\ldots$ Since $k$ must be an integer (representing the number of terms), we take the greatest integer less than or equal to $14.714\ldots$, which is $14$. Therefore, there are $14$ terms common to both A.P.s.

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Common Mistakes & Tips

• Finding the First Common Term: Listing out the first few terms is efficient for small numbers. For larger numbers, setting the general terms equal and solving the resulting linear Diophantine equation for the smallest positive integer solutions is a more robust method.
• Common Difference of the New A.P.: Always use the LCM of the common differences of the original A.P.s. Do not use the GCD or simply the product if the common differences are not coprime.
• Upper Bound: Ensure you use the smaller of the two last terms as the upper limit for the common terms. Using the larger last term would incorrectly include terms that are not present in the shorter of the two original sequences.
• Integer Value for Number of Terms: When solving for $k$, remember that the number of terms must be a positive integer. Always take the floor of the resulting decimal value for $k$.

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Summary To find the number of common terms between two arithmetic progressions, we first identify the first term and common difference of each A.P. The common terms themselves form a new A.P. whose first term is the smallest common value found in both original sequences, and whose common difference is the LCM of the original common differences. The number of such common terms is then determined by finding how many terms of this new A.P. are less than or equal to the minimum of the last terms of the original A.P.s. In this case, the first common term is $23$, the common difference of the new A.P. is $28$, and the upper bound is $407$. Solving for the number of terms yields $14$.

The final answer is $\boxed{14}$.