5. Consider the following linear pattems: Pattem $$ A: 88 ; 84 ; 80 ; 76 ; 72 ; \ldots $$ Pattern $$ B: 5 ; 10 ; 15 ; 20 ; 25 ; \ldots $$ If the pathems are continued, how many numbers would appear in both pattems? $$ \begin{array}{lll} ext { (A) } 0 & ext { (B) } 2 & ext { (D) } 8\end{array} $$ (D) 8

Question

5. Consider the following linear pattems: Pattem $$ A: 88 ; 84 ; 80 ; 76 ; 72 ; \ldots $$ Pattern $$ B: 5 ; 10 ; 15 ; 20 ; 25 ; \ldots $$ If the pathems are continued, how many numbers would appear in both pattems? $$ \begin{array}{lll}	ext { (A) } 0 & 	ext { (B) } 2 & 	ext { (D) } 8\end{array} $$ (D) 8

UpStudy AI · Accepted Answer

To determine how many numbers appear in both patterns $$ A $$ and $$ B $$, we first need to identify the general formulas for each pattern.

### Pattern A:
The sequence is $$ 88, 84, 80, 76, 72, \ldots $$. This is an arithmetic sequence where:
- The first term $$ a_1 = 88 $$
- The common difference $$ d = -4 $$

The $$ n $$-th term of an arithmetic sequence can be expressed as:
$$
a_n = a_1 + (n-1) \cdot d
$$
Substituting the values:
$$
a_n = 88 + (n-1)(-4) = 88 - 4(n-1) = 92 - 4n
$$

### Pattern B:
The sequence is $$ 5, 10, 15, 20, 25, \ldots $$. This is also an arithmetic sequence where:
- The first term $$ b_1 = 5 $$
- The common difference $$ d = 5 $$

The $$ m $$-th term of this sequence can be expressed as:
$$
b_m = b_1 + (m-1) \cdot d
$$
Substituting the values:
$$
b_m = 5 + (m-1)(5) = 5m
$$

### Finding Common Terms:
We need to find values of $$ n $$ and $$ m $$ such that:
$$
92 - 4n = 5m
$$
Rearranging gives:
$$
4n + 5m = 92
$$

### Solving the Equation:
We can express $$ m $$ in terms of $$ n $$:
$$
5m = 92 - 4n \implies m = \frac{92 - 4n}{5}
$$
For $$ m $$ to be an integer, $$ 92 - 4n $$ must be divisible by $$ 5 $$.

Calculating $$ 92 \mod 5 $$:
$$
92 \div 5 = 18 \quad \text{remainder } 2 \quad \Rightarrow \quad 92 \equiv 2 \mod 5
$$
Thus, we need:
$$
4n \equiv 2 \mod 5
$$
To solve for $$ n $$, we can find the multiplicative inverse of $$ 4 \mod 5 $$. The inverse is $$ 4 $$ since:
$$
4 \cdot 4 \equiv 1 \mod 5
$$
Multiplying both sides of $$ 4n \equiv 2 $$ by $$ 4 $$:
$$
16n \equiv 8 \mod 5 \implies n \equiv 3 \mod 5
$$
This means:
$$
n = 5k + 3 \quad \text{for integers } k
$$

### Finding Corresponding $$ m $$:
Substituting $$ n $$ back into the equation for $$ m $$:
$$
m = \frac{92 - 4(5k + 3)}{5} = \frac{92 - 20k - 12}{5} = \frac{80 - 20k}{5} = 16 - 4k
$$

### Finding Valid $$ n $$ and $$ m $$:
Both $$ n $$ and $$ m $$ must be positive integers:
1. For $$ n = 5k + 3 $$ to be positive, $$ k \geq 0 $$.
2. For $$ m = 16 - 4k $$ to be positive, $$ 16 - 4k > 0 \implies k < 4 $$.

Thus, $$ k $$ can take values $$ 0, 1, 2, 3 $$:
- $$ k = 0 $$: $$ n = 3, m = 16 $$
- $$ k = 1 $$: $$ n = 8, m = 12 $$
- $$ k = 2 $$: $$ n = 13, m = 8 $$
- $$ k = 3 $$: $$ n = 18, m = 4 $$

### Conclusion:
The valid pairs $$ (n, m) $$ are:
- $$ (3, 16) $$
- $$ (8, 12) $$
- $$ (13, 8) $$
- $$ (18, 4) $$

Thus, there are **4 pairs** of $$ (n, m) $$ that satisfy the equation, meaning there are **4 common numbers** in both patterns.

The answer is not among the options provided, but based on the calculations, the correct number of common terms is **4**.
There are 4 numbers that appear in both patterns A and B.

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