Prime Factorization, Squares, and Cubes
Prime factorize 126.
2 x 32 x 7
Find the GCF and LCM of 24 and 36.
GCF = 12, LCM = 72
Define range.
The difference of the largest number and the smallest number in a set
The least common denominator of 1/3 and 3/4 is:
12
The reciprocal of 1 is:
1
(ab)2n because it is a product of two perfect squares.
Alex works every 6 days, and Lizzie works every 4 days. In how many days will they work together again, if they worked together today?
12
Find the mean of 22, 23, and 24.
28/3
A watermelon weighs 1.125 kg. What is the weight of one watermelon and a bottle of water in kilograms, if the water bottle weighs 250 g? (in fraction form)
1 3/8 kg
Grogg bikes 20 km a week. On average, how many kilometres does he bike per day?
20/7 km or 2 6/7 km
Grogg is thinking of a perfect cube. If he multiplies the number by 27, it becomes a perfect square and a perfect cube. His number is less than 100. What is his number?
27
Prove that any two consecutive integers are relatively prime, without giving any examples.
Their difference is always 1. Therefore, the GCF is 1 due to the Euclidean Algorithm.
Find the median of the set {a, b, c, d, e, f, g, h} where a > 0 and the variables are in increasing order.
(d+e)/2
Lizzie is serving wine to three customers. She has one full cup of wine. She gives one half of the wine to the first customer, then one half of the remaining to the second customer, then one half of the remaining to the third customer. What fraction of her cup is empty?
15/16
Grogg and Winnie are racing. Winnie runs 20% more laps than Grogg. If Grogg ran 8 1/3 laps, how many laps did Winnie run?
10
Explain why a perfect square has an odd number of factors. Do NOT give examples.
Every number has at least one factor pair, or at least two factors. However, perfect squares have a factor pair of two equal factors. Therefore, that pair only counts as one factor instead of two.
Winnie is thinking of three natural numbers, a, b, and c, where a < b < c. She finds out that a is a factor of 2b, and b is a factor of 3c. What is their LCM?
6c
Alex, Lizzie, Winnie, and Grogg are playing a game of basketball. Alex scored 25 points, Winnie scored three more points than Grogg, Lizzie scored 32 points, and the players scored an average of 27 points. How many points did Winnie score?
27
Alex is doing a science experiment. He has a 40 mL solution with 5/8 of salt. He adds a 25 mL solution with 3/5 of salt. What is the concentration of salt in the combined solution?
8/13
A ball bounces to 4/5 of its original height. If it is dropped from a height of 5 metres, how many bounces will it take for it to bounce lower than a metre?
6
What is the smallest number that must be multiplied to 7! to make a perfect square and perfect cube at the same time? Leave your answer in simplified exponent form.
22 x 34 x 55 x 75
Winnie and Lizzie have entered an escape room. The code that will help them escape is made up of four nonzero digits. The first two digits are distinct, with the first digit being less than the second digit. The last two digits of the code, combined with each other, is the LCM of the first two digits of the code. For example, 8 and 9 combined with each other is 89. Find all the possible combinations of the code.
2714, 2918, 3412, 3515, 3721, 3824, 4520, 4612, 4728, 4936, 5630, 5735, 5840, 5945, 6742, 6824, 6918, 7856, 7963, 8972
Alex is thinking of a six-number set where the mean, median, and range are all the same. A number can only repeat twice. All numbers are less than or equal to 5. Name all the possible combinations.
{1, 1, 2, 2, 3, 3}
Alex decides to walk back and forth from his house. He walks 2 1/2 km south, then 4 5/6 km north, then 7 3/4 km south, and then 4 1/3 km north. How far is he from his house now?
1 1/12 km
Lizzie is building a five-storey tower. Each floor has a weight that is less than 10 kg, and the weight is a whole number. The weight of one floor cannot be greater than or equal to the floor above it, or else the tower will collapse. The weights of the floor are arranged into a big fraction (1/a)/b/(1/c)/d/(1/e). Find the value of the fraction when weights of 8 kg, 2 kg, 5 kg, 3 kg, and 7 kg are used.
20/21