Find the number of occurrences

Some definitions - note that the "bi-" frequencies are a bit ambiguous:

• Weekly = 1/week, i.e. something is done once every week

• Biweekly = 1/(2 weeks)

• Quarterly = 1/(3 months)

• Semimonthly = 2/month

• Bimonthly =  1/(2 months)

• Annually = 1/year

• Monthly = 1/month

"Biweekly" is sometimes used to mean "twice per week", but I've looked up "semimonthly" and it seems to strictly mean "once per half-month" or "twice per month". So to avoid ambiguity, I'll take "bimonthly" and "biweekly" to mean "once every two months" and "once every two weeks".

Some conversion factors:

• 1 year = 12 months = 52 weeks

1. Weekly for 1 year:

1/week • 52 weeks = 52 times

2. Biweekly for 2 years:

1/(2 weeks) • 104 weeks = 52 times

3. Quarterly for 4 years:

1/(3 months) • 48 months = 16 times

4. Semimonthly for 1 year:

2/month • 12 months = 24 times

5. Bimonthly for 2 years:

1/(2 months) • 24 months = 12 times

6. Annually for 25 years:

1/year • 25 years = 25 times

7. Monthly for 1/2 year:

1/month • 6 months = 6 times

8. Weekly for 1.5 years:

1/week • 78 weeks = 78 times

9. Semimonthly for 2.5 years:

2/month • 30 months = 60 times

the discriminant is 64 + 4*1*12 = 112

this is prime.

step-by-step explanation: