Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$
There are five prime numbers between $4$ and $18:$ namely $5,$ $7,$ $11,$ $13,$ and $17.$ Hence the product of any two of these is odd and the sum is even. Because $$xy-(x+y)=(x-1)(y-1)-1$$increases as either $x$ or $y$ increases (since both $x$ and $y$ are bigger than $1$), the answer must be an odd number that is no smaller than $$23=5\cdot 7-(5+7)$$and no larger than $$191=13\cdot 17-(13+17).$$The only possibility among the options is $\boxed{119},$ and indeed $119=11\cdot 13-(11+13).$
119
Level 2
Number Theory
Problem: Two different prime numbers between $4$ and $18$ are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained? $$
\text A. \ \ 21 \qquad \text B. \ \ 60 \qquad \text C. \ \ 119 \qquad \text D. \ \ 180 \qquad \text E. \ \ 231
$$
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, we first identify the prime numbers between 4 and 18. These primes are 5, 7, 11, 13, 17. 

Next, we denote the two chosen prime numbers as \( p_1 \) and \( p_2 \). The expression we need to evaluate is \( p_1 p_2 - (p_1 + p_2) \). This can be rewritten as:

\[
p_1 p_2 - (p_1 + p_2) = p_1 p_2 - p_1 - p_2
\]

Factoring this expression gives:

\[
p_1 p_2 - p_1 - p_2 = (p_1 - 1)(p_2 - 1) - 1
\]

This shows that the result is one less than the product of \( (p_1 - 1) \) and \( (p_2 - 1) \). 

Next, we calculate \( (p_1 - 1)(p_2 - 1) \) for all pairs of the identified prime numbers:

1. For \( p_1 = 5 \) and \( p_2 = 7 \):
   \[
   (5 - 1)(7 - 1) = 4 \cdot 6 = 24 \quad \Rightarrow \quad 24 - 1 = 23
   \]

2. For \( p_1 = 5 \) and \( p_2 = 11 \):
   \[
   (5 - 1)(11 - 1) = 4 \cdot 10 = 40 \quad \Rightarrow \quad 40 - 1 = 39
   \]

3. For \( p_1 = 5 \) and \( p_2 = 13 \):
   \[
   (5 - 1)(13 - 1) = 4 \cdot 12 = 48 \quad \Rightarrow \quad 48 - 1 = 47
   \]

4. For \( p_1 = 5 \) and \( p_2 = 17 \):
   \[
   (5 - 1)(17 - 1) = 4 \cdot 16 = 64 \quad \Rightarrow \quad 64 - 1 = 63
   \]

5. For \( p_1 = 7 \) and \( p_2 = 11 \):
   \[
   (7 - 1)(11 - 1) = 6 \cdot 10 = 60 \quad \Rightarrow \quad 60 - 1 = 59
   \]

6. For \( p_1 = 7 \) and \( p_2 = 13 \):
   \[
   (7 - 1)(13 - 1) = 6 \cdot 12 = 72 \quad \Rightarrow \quad 72 - 1 = 71
   \]

7. For \( p_1 = 7 \) and \( p_2 = 17 \):
   \[
   (7 - 1)(17 - 1) = 6 \cdot 16 = 96 \quad \Rightarrow \quad 96 - 1 = 95
   \]

8. For \( p_1 = 11 \) and \( p_2 = 13 \):
   \[
   (11 - 1)(13 - 1) = 10 \cdot 12 = 120 \quad \Rightarrow \quad 120 - 1 = 119
   \]

9. For \( p_1 = 11 \) and \( p_2 = 17 \):
   \[
   (11 - 1)(17 - 1) = 10 \cdot 16 = 160 \quad \Rightarrow \quad 160 - 1 = 159
   \]

10. For \( p_1 = 13 \) and \( p_2 = 17 \):
    \[
    (13 - 1)(17 - 1) = 12 \cdot 16 = 192 \quad \Rightarrow \quad 192 - 1 = 191
    \]

After evaluating all pairs, the only result that matches one of the options provided is \( 119 \). Thus, the final answer is \( 119 \).<|END_OF_LATENT|> The final answer is $119$.