Primes
This is the only even prime number.
2
Give the full prime factorization of 60.
2² × 3 × 5
True or false: every prime number greater than 2 is odd.
True
Find the GCF of 24 and 36.
12
How many total divisors does 36 have?
9: 1, 2, 3, 4, 6, 9, 12, 18, 36
Of 51 and 53, only one is prime. Which one, and why isn't the other?
53, 17x3=51
How many distinct prime factors does 210 have?
4 (2, 3, 5, and 7)
Primes are defined as having exactly two divisors. Explain why 1 doesn't count.
1 has only one divisor (itself), not two, so it fails the definition.
Find the LCM of 8 and 12.
24
A rectangular garden has an area of 84 sq ft. Both whole-number sides exceed 2 ft, and length ≠ width. Give one valid pair of dimensions.
For example, 6 × 14 (7 × 12 and 4 × 21 also work)
Find the twin prime pair (two primes differing by exactly 2) that lies between 40 and 45.
41 and 43
What is the smallest positive integer with exactly 3 distinct prime factors?
30 (2 × 3 × 5)
Name the next prime number after 97.
101
Two numbers have a GCF of 6 and an LCM of 72. One of them is 24. Find the other.
18, since GCF × LCM = product of the numbers: 6 × 72 ÷ 24 = 18
48 cookies must be split evenly into more than 1 box, with more than 1 cookie per box. What's the largest possible number of cookies per box?
24 cookies per box (2 boxes)
Find the only set of three primes of the form n, n+2, n+4 (three numbers each 2 apart).
3, 5, and 7
A number's prime factorization is 2³ × 3². How many total divisors does it have?
12, since (3+1)(2+1) = 12
Find the smallest number greater than 1 that leaves a remainder of 1 when divided by 2, 3, 5, and 7.
211, one more than 2×3×5×7 = 210
Two numbers have a GCF of 8 and an LCM of 96. One number is 32 — find the other.
24, since 8 × 96 ÷ 32 = 24
A number under 100 has exactly 8 factors and exactly 2 distinct prime factors. What's the largest such number?
88 = 2³ × 11, which has (3+1)(1+1) = 8 factors
Explain why 3, 5, 7 is the only prime triplet of that form that will ever exist.
Among any three numbers spaced 2 apart, one is always a multiple of 3. Unless that multiple of 3 is 3 itself, it can't be prime so every other attempt fails.
Find the smallest positive integer with exactly 12 divisors.
60 = 2² × 3 × 5, giving (2+1)(1+1)(1+1) = 12 divisors
Write 100 as a sum of two primes, in two different ways.
For example, 3 + 97 and 47 + 53 (several correct pairs exist)
Explain why no number N can satisfy both GCF(N, 18) = 6 and GCF(N, 30) = 15 at the same time.
GCF with 18 = 6 forces N to be even. GCF with 30 = 15 forces N to be odd, since 15 has no factor of 2 even though 30 does. N can't be both.
A locker code is the product of three different primes, each less than 20. Find the three primes that make the product as large as possible without going over 1000.
3, 17, and 19: their product is 969, the closest you can get to 1000 without exceeding it