Homework Packet
1. Lowest-Common-Multiples
2. Divisors
3. Remainders
4. Greatest Common Factors
5. Pythagorean Theorem
6. Proportions
7. Percents with Triangles
Lowest-Common-Multiple (LCM)
To find the LCM of a set of numbers, look for where the numbers intersect.
Example 1 — What is the LCM of 4 and 10?
Multiples of 4: 4, 8, 12, 16, 20
Multiples of 10: 10, 20, 30, 40, 50
Basically, where do multiples of 4 and multiples of 10 hit the same number?
Answer: 20
Example 2 — What is the LCM of 2, 3, 5?
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
3 6 9 12 15 18 21 24 27 30
5 10 15 20 25 30
Answer: 30
Strategies
● For questions that have x and y or unknown numbers, it is very helpful to write out the
multiples to see what the possible numbers could be.
Questions:
1. Find the LCM of 5 and 7.
2. Find the LCM of 11 and 22.
3. Find the LCM of 50 and 2.
4. Find the LCM of 3, 5, and 10.
5. Find the LCM of 100 and 1000.
6. Find the LCM of 1, 2, 3, 4, 5.
7. 6 and X have a LCM of 30. What number could X be?
8. A and B have a LCM of 6.
a. What number could A be?
b. What number could B be?
Divisors
2 times 3 = 6.
That means 2 and 3 are divisors of 6.
5 is not a divisor of 6 because there will be a remainder.
“Factors” and “divisors” mean the same thing.
The factors of 6 are 1, 2, 3, 6.
The divisors of 6 are also 1, 2, 3, 6.
Strategies:
● The most common strategy is dividing a number by 2, 3, 5, 7, or 9. If it works, keep going,
until you can’t break it down any more.
Questions:
1. Find the divisors for 35.
2. Find the divisors for 350.
3. How many more divisors does 24 have than 12?
4. How many divisors do 3 and 6 have in common?
5. Which number has the most divisors?
A. 12
B. 100
C. 30
D. 50?
6.
1 has 1 divisors
10 has 4 divisors
100 has 9 divisors
1000 has 16 divisors
1*1=1
2*2=4
3*3=9
4 * 4 = 16
Follow the pattern: How many divisors will 10,000 have?
7. Find all the divisors of 28 and add them up. What does it equal?
Remainders
Notes:
● Remainder questions are often paired with divisor questions.
● A very typical type of question is:
The remainder of A / 7 is 3. What number could A be?
● To solve this question, guess-and-check a few times. Could A be 14?
● 14 / 7 = 2. Remainder 0
● Could A be 15? 15 / 7 is 2 remainder 1. Getting closer.
● 16 / 7 is 2 remainder 2
● 17 /7 is 2 remainder 3. The answer is A could be 17.
Questions
1. The remainder of A / 7 is 6. A is bigger than 30 and less than 40. What number could A be?
2. Dividing 37 by a certain number gives a remainder of 1. What could this number be?
3. Dividing 37 by a number gives a remainder of 1. If you divide 38 by this number, what
remainder will you get?
4. What is the remainder when 83 is divided by 82?
5. What is the remainder when 83 is divided by 12?
6. How much greater is the remainder of 83 / 82 than the remainder of 83 / 12?
Greatest Common Factor (GCF)
To find the GCF among a series of numbers, look at the factors for each of the numbers, and find
the LARGEST one that they all have.
Factors of 6: 1, 2, 3, 6
Factors of 8: 1,2,4, 8
The GCF of 6 and 8 is 2.
Strategies:
● Write out the factors for the number and then circle the ones that are shared in common.
● The greatest common factor will be the largest one.
Questions:
1. Find the GCF of 11 and 12.
2. Find the GCF of 12 and 24.
3. Find the GCF of 10, 15, 20.
4. Find the GCF of 10, 15, 20, and 25.
5. 6 and X have a GCF of 3. What number could X be?
6. 15 and Y have a GCF of 5. What number could Y be?
Proportions
Proportion questions are very similar to finding “equivalent fractions.”
● 5 apples for 8 dollars.
● How many apples can you buy with 40 dollars?
● The equivalent proportion is 25/40.
● You can buy 25 apples.
Questions:
1. 7 to 9 is the same as 14 to ?
2. 4 to 12 is the same 40 to ?
3. 3A to 7 is the same as 21A to X. What is X?
4. 8X to 3Y is the same as 24X to how many Y?
5. Sarah sells cakes 10 for 2 dollars, and Sam sells them 15 for 3 dollars.
A. Sarah gives a better deal
B. Sam gives a better deal
C. Sarah and Sam give the same deal
6. 1 dollar equals 4 quarters. 1 dollar also equals 10 dimes. What is the simplified proportion of
quarters to dimes in a single dollar?
Percents with Triangles
Notes:
● The angles of a triangle always add up to 180 degrees.
● Isosceles means 2 of the angles are the same.
● Scalene means none of the angles are the same.
● Equilateral means all of the angles are 60 degrees.
Examples:
● Say for example angle1 is 40% longer than angle2 and angle 2 is 40 degrees. 40% of 40 is 16,
add that on top, and you get angle1 is 56 degrees.
● You know the entire triangle adds up to 180 degrees.
● That means angle3 is 180 - 40 - 56 which equals 84 degrees.
Questions:
1. In a triangle, angle X is 20% larger than angle Y which is 30 degrees. How big is angle Y?
2. In a triangle, the first angle is 30, the second angle is 36. What is the third angle?
3. In a triangle, angle X is 20% larger than angle Y which is 30 degrees. How big is the third
angle, angle Z?
4. In an isosceles triangle, two of the angles are 50 and 80. What is the third angle?
5. In a scalene triangle, Angle1 = 40, Angle2 = 30% more than Angle1, how much is angle 3?
6. In a scalene triangle, Angle1 = 40, Angle2 = 30% more than Angle1. What is the ratio of
angle 3 to angle 1? (Write the ratio fraction in simplified form)
