You can find many official and unofficial resources for past problems and solutions, but remember that copyright restrictions apply to official material.
A number with exactly 4 positive factors is either:
Use estimation and mental shortcuts to avoid time-consuming long-hand arithmetic. Pattern Recognition: Mathcounts National Sprint Round Problems And Solutions
13S=19+227+381+…one-third cap S equals one-nineth plus 2 over 27 end-fraction plus 3 over 81 end-fraction plus …
is a positive integer. Thus, both factors must be positive integers. The number of ordered pairs You can find many official and unofficial resources
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Because the die is fair, look at the transitions when the game does not end (rolling a 1, 2, 3, 4, or 5). There is a 15one-fifth chance of rolling a 3, which keeps you in the same state. There is a 25two-fifths chance of rolling a 1 or 4, moving you to the next state. There is a 25two-fifths chance of rolling a 2 or 5, moving you to the other state. Thus, both factors must be positive integers
10!=28×34×52×7110 exclamation mark equals 2 to the eighth power cross 3 to the fourth power cross 5 squared cross 7 to the first power