To solve for the number of integers \( n \) within the interval \( 70 \leq n \leq 90 \) that can be expressed in the form
\[ n = ab + 2a + 3b, \]
we can rearrange the equation for \( n \):
\[ n = ab + 2a + 3b = a(b + 2) + 3b. \]
Now, we can let \( m = b + 2 \). Then, we can rewrite \( b \) in terms of \( m \) as \( b = m - 2 \). Substituting this back into our formula gives:
\[ n = a(m) + 3(m - 2) = am + 3m - 6 = (a + 3)m - 6. \]
Now, we can rearrange this to isolate \( m \):
\[ n + 6 = (a + 3)m \implies m = \frac{n + 6}{a + 3}. \]
Since \( m \) is a positive integer, \( n + 6 \) must be divisible by \( a + 3 \). We can explore which integers give permissible \( m \) values by determining values of \( n + 6 \):
- The smallest \( n \) is \( 70 \), so \( n + 6 = 76 \).
- The largest \( n \) is \( 90 \), so \( n + 6 = 96 \).
Consequently, we need to examine the integers from \( 76 \) to \( 96 \):
\[ 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96. \]
Next, we can determine the possible values of \( a + 3 \). Since \( a \) is a positive integer, \( a \geq 1 \) implies \( a + 3 \geq 4 \). Therefore, \( m \) can take any integer value for \( a + 3 \in {4, 5, 6, \ldots}\).
To find integers \( n \), we need \( n + 6 \) to be divisible by each \( a + 3 \):
-
For \( k = 4 \): \( n + 6 = 76 \), \( n \equiv 0 \mod 4 \) gives \( n = 70, 74, 78, 82, 86, 90 \).
-
For \( k = 5 \): \( n + 6 = 76 \equiv 1 \mod 5 \) means \( n \equiv -1 \mod 5 \), giving \( n = 74, 79, 84, 89 \).
-
For \( k = 6 \): \( n + 6 = 76 \equiv 4 \mod 6 \) means \( n \equiv 2 \mod 6 \), yielding \( n = 72, 78, 84, 90 \).
-
For \( k = 7 \): \( n + 6 = 76 \equiv 6 \mod 7 \) which gives \( n \equiv 1 \mod 7 \) as \( n = 70, 77, 84, 91 \).
-
For \( k = 8 \): \( n + 6 = 76 \equiv 4 \mod 8 \) gives \( n \equiv 2 \mod 8 \), yielding \( n = 70, 78, 86 \).
-
For \( k = 9 \): \( n + 6 = 76 \equiv 4 \mod 9 \) gives \( n \equiv 5 \mod 9 \), yielding \( n = 74, 83, 92 \).
Now, we can collect all the possible \( n \):
From \( k = 4 \): \( 70, 74, 78, 82, 86, 90 \)
From \( k = 5 \): \( 74, 79, 84, 89 \)
From \( k = 6 \): \( 72, 78, 84, 90 \)
From \( k = 7 \): \( 70, 77, 84, 91 \)
From \( k = 8 \): \( 70, 78, 86 \)
From \( k = 9 \): \( 74, 83, 92 \)
Next, let's find unique \( n \) from all these lists:
- Compiling unique values from the sets we found: \( 70, 72, 74, 77, 78, 79, 82, 83, 84, 86, 89, 90, 91, 92 \).
Thus, the unique values of \( n \) between \( 70 \leq n \leq 90 \) are:
\[ 70, 72, 74, 77, 78, 79, 82, 83, 84, 86, 89, 90. \]
Counting these gives us:
\[ 70, 72, 74, 77, 78, 79, 82, 83, 84, 86, 89, 90 \Rightarrow \text{ 12 integers. } \]
Therefore, the number of integers \( n \) that can be expressed in the desired form is \( \boxed{12} \).
1 answer