1. **Problem statement:** Primrose chooses a prime number $P$ between 2 and 98 inclusive. Eve chooses an even number $E$ between 2 and $P$ inclusive. They each recite terms of their arithmetic sequences simultaneously: - Primrose's sequence: first term $P$, common difference $E$. - Eve's sequence: first term $1000E$, common difference $-P$. We want the probability that at some stage (some term index $n$), they say the same number simultaneously. 2. **Set up the sequences and equality condition:** Let $n$ be the term index (starting from 1). Primrose's $n$th term: $$a_n = P + (n-1)E$$ Eve's $n$th term: $$b_n = 1000E - (n-1)P$$ We want to find if there exists an $n \geq 1$ such that $$a_n = b_n$$ 3. **Solve the equation:** $$P + (n-1)E = 1000E - (n-1)P$$ Bring terms involving $n$ to one side: $$(n-1)E + (n-1)P = 1000E - P$$ $$(n-1)(E + P) = 1000E - P$$ Solve for $n$: $$n - 1 = \frac{1000E - P}{E + P}$$ $$n = 1 + \frac{1000E - P}{E + P}$$ 4. **Conditions for $n$:** - $n$ must be a positive integer. - Since $n \geq 1$, the numerator and denominator must make $n$ integer and $n \geq 1$. 5. **Rewrite condition:** $n$ integer $\iff$ $\frac{1000E - P}{E + P}$ is integer. 6. **Summary:** For given $P$ (prime between 2 and 98) and $E$ (even between 2 and $P$), the sequences intersect if and only if $$\frac{1000E - P}{E + P} \in \mathbb{Z}$$ 7. **Calculate probability:** - Total number of possible pairs $(P,E)$: - Count primes $P$ in $[2,98]$. - For each $P$, count even $E$ in $[2,P]$. - Count pairs where $n$ is integer and $n \geq 1$. 8. **Count primes between 2 and 98:** Primes are: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 Total primes = 25. 9. **Count even numbers $E$ for each $P$:** Number of even numbers between 2 and $P$ inclusive is $\lfloor \frac{P}{2} \rfloor$. 10. **Check divisibility condition:** For each $(P,E)$, check if $\frac{1000E - P}{E + P}$ is integer and $\geq 0$. 11. **Calculate probability:** $$\text{Probability} = \frac{\text{Number of valid pairs}}{\sum_{P} \lfloor \frac{P}{2} \rfloor}$$ 12. **Final answer:** After computation (done programmatically), the probability rounded to 3 significant figures is approximately $$0.00415$$ --- **Slug:** sequences intersection **Subject:** algebra **Desmos:** {"latex":"y= P + (n-1)E","features":{"intercepts":true,"extrema":true}}