How it works
Every whole number greater than 1 can be written as a product of primes in exactly one way (apart from order). Build it with a factor tree, dividing by small primes first: 360 = 2 × 180 = 2 × 2 × 90 = … = 2³ × 3² × 5.
Once you have the factorization, many questions answer themselves. To raise a number to a power, multiply every exponent: 12⁵ = (2² × 3)⁵ = 2¹⁰ × 3⁵. To find how many times a prime p divides n!, add the multiples of p, p², p³, … up to n:
⌊n/p⌋ + ⌊n/p²⌋ + ⌊n/p³⌋ + …
Prime Factorization practice problems with solutions
These problems come from Mathyessy's question bank. Try each one before opening the solution.
How many prime factors does 360 have counting multiplicity?
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Answer: 6
360 = 2³ × 3² × 5, so counting repeats there are 3 + 2 + 1 = 6 prime factors.
If 12 to the power 5 equals 2 to the power a times 3 to the power b, what is a plus b?
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Answer: 15
12 = 2² × 3, so 12⁵ = 2¹⁰ × 3⁵. Then a + b = 10 + 5 = 15.
What is the exponent of 3 in the prime factorization of 20 factorial?
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Answer: 8
Multiples of 3 up to 20: ⌊20/3⌋ = 6. Multiples of 9 give an extra 3 each: ⌊20/9⌋ = 2. 27 is past 20. Total 6 + 2 = 8.
If n equals 2 to the power 5 times 3 squared times 7, how many even divisors does n have?
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Answer: 30
All divisors: (5 + 1)(2 + 1)(1 + 1) = 36. Odd divisors use no 2: (2 + 1)(1 + 1) = 6. Even divisors: 36 − 6 = 30.
Keep going
Related topics
- Counting divisors
- Greatest common divisor (GCD)
- Least common multiple (LCM)
- Trailing zeros in factorials
Related practice sets in the app
All Number Theory topics · Contest math practice by topic · MATHCOUNTS practice · AMC 8 practice