Set Definitions
Set A contains the first 101 terms of an arithmetic progression (A.P.) with first term $a_1 = 1$ and common difference $d_1 = 5$. The general term is $T_A(n) = a_1 + (n-1)d_1 = 1 + (n-1)5 = 5n - 4$, for $1 \le n \le 101$. The maximum value in A is $T_A(101) = 5(101) - 4 = 505 - 4 = 501$.
Set B contains the first 71 terms of an A.P. with first term $a'_1 = 9$ and common difference $d'_2 = 7$. The general term is $T_B(m) = a'_1 + (m-1)d'_2 = 9 + (m-1)7 = 7m + 2$, for $1 \le m \le 71$. The maximum value in B is $T_B(71) = 7(71) + 2 = 497 + 2 = 499$.
Terms Divisible by 3 in Set A
A term $T_A(n) = 5n - 4$ is divisible by 3 if:
$5n - 4 \equiv 0 \pmod{3}$
$2n - 1 \equiv 0 \pmod{3}$
$2n \equiv 1 \pmod{3}$
Multiplying by 2 (the inverse of 2 mod 3):
$4n \equiv 2 \pmod{3}$
$n \equiv 2 \pmod{3}$
So, terms in A divisible by 3 occur when $n$ is of the form $3k - 1$ (for $k=1, 2, ...$). Let's check the range: $n = 2, 5, 8, ...$. These terms form an A.P. $T_A(2) = 5(2)-4 = 6$. The common difference for this subsequence is $3 \times d_1 = 3 \times 5 = 15$. The sequence is $6, 21, 36, ...$.
Terms Divisible by 3 in Set B
A term $T_B(m) = 7m + 2$ is divisible by 3 if:
$7m + 2 \equiv 0 \pmod{3}$
$m + 2 \equiv 0 \pmod{3}$
$m \equiv -2 \equiv 1 \pmod{3}$
So, terms in B divisible by 3 occur when $m$ is of the form $3j - 2$ (for $j=1, 2, ...$). Let's check the range: $m = 1, 4, 7, ...$. These terms form an A.P. $T_B(1) = 7(1)+2 = 9$. The common difference for this subsequence is $3 \times d'_2 = 3 \times 7 = 21$. The sequence is $9, 30, 51, ...$.
Intersection of Sets A and B Divisible by 3
We need to find the common terms in the two subsequences derived above:
- Subsequence from A (divisible by 3): $6, 21, 36, 51, 66, 81, 96, ...$ (A.P. with first term 6, common difference 15).
- Subsequence from B (divisible by 3): $9, 30, 51, 72, 93, 114, ...$ (A.P. with first term 9, common difference 21).
The common terms will form an A.P. with a common difference equal to the LCM of 15 and 21.
$LCM(15, 21) = LCM(3 \times 5, 3 \times 7) = 3 \times 5 \times 7 = 105$.
The first common term is 51.
The sequence of common terms (in $A \cap B$ and divisible by 3) starts with 51 and has a common difference of 105. The general term is $51 + k \times 105$, where $k \ge 0$.
Counting Common Elements
These common terms must be within the bounds of both Set A and Set B. The maximum possible value is $\min(501, 499) = 499$.
We need to find the number of terms $k$ such that:
$51 + k \times 105 \le 499$
$k \times 105 \le 499 - 51$
$k \times 105 \le 448$
$k \le \frac{448}{105} \approx 4.267$
Since $k$ must be an integer, the maximum value for $k$ is 4.
The possible values for $k$ are $0, 1, 2, 3, 4$. This gives a total of $4 + 1 = 5$ terms.
The terms are: 51, 156, 261, 366, 471.
Therefore, the number of elements in $A \cap B$ which are divisible by 3 is 5.
Similar Questions
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
- Let $a_1 = 1$ and for $n \ge 1, a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2(n+1)^2}$. Then $\left| \sum_{n=1}^\infty \left( a_n - \frac{2}{n^2} \right) \right|$ is equal to ____________.
If $ \left(\frac{1}{^{15}C_0} + \frac{1}{^{15}C_1}\right)$ $\left(\frac{1}{^{15}C_1} + \frac{1}{^{15}C_2}\right)$ $\dots$ $ \left(\frac{1}{^{15}C_{12}} + \frac{1}{^{15}C_{13}}\right)$ = $\frac{\alpha^{13}}{^{14}C_0 ^{14}C_1 \dots ^{14}C_{12}} $, then $30\alpha$ is equal to __________.
- Three persons enter in a lift at the ground floor. The lift will go upto 10$^{\text{th}}$ floor. The number of ways, in which the three persons can exit the lift at three different floors, if the lift does not stop at first, second and third floors, is equal to _________.
- The number of 4-letter words, with or without meaning, which can be formed using the letters PQR PQRSTUVP, is ___________.
- Let $|A| = 6$, where A is a $3 \times 3$ matrix. If $|\text{adj}(3\text{adj}(A^2 \cdot \text{adj}(2A)))| = 2^m \cdot 3^n$, $m, n \in \mathbf{N}$, then $m + n$ is equal to ___________.
Important Questions from Algebra
Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.
- The number of $3 \times 2$ matrices A, which can be formed using the elements of the set $\{-2, -1, 0, 1, 2\}$ such that the sum of all the diagonal elements of $A^T A$ is 5, is ______
- The number of numbers greater than 5000, less than 9000 and divisible by 3, that can be formed using the digits 0, 1, 2, 5, 9, if the repetition of the digits is allowed, is ______
- Let $z = (1 + i)(1 + 2i)(1 + 3i) \dots (1 + ni)$, where $i = \sqrt{-1}$. If $|z|^2 = 44200$, then $n$ is equal to _______
- Let $S = \{(m, n) : m, n \in \{1, 2, 3, \dots, 50\}\}$. If the number of elements $(m, n)$ in S such that $6^m + 9^n$ is a multiple of 5 is p and the number of elements $(m, n)$ in S such that $m + n$ is a square of a prime number is q, then $p + q$ is equal to ____________.
