All the solutions provided in McGraw Hill Math Grade 5 Answer Key PDF Chapter 8 Lesson 5 Least Common Multiple will give you a clear idea of the concepts.
McGraw-Hill My Math Grade 5 Answer Key Chapter 8 Lesson 5 Least Common Multiple
Math in My World
Example 1
Tom hits golf balls at the driving range every 3 days, practices putting every 4 days, and goes golfing every 6 days. If he did all three activities today, in how many days will he do all three activities again?
Find and circle the LCM of 3, 4 and 6 by listing nonzero multiples of each number.
3: 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30.
4: 4, 8, 12, 26, 20, 24, 28, 32, 36 and 40.
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.
So, Tom will complete all three activities again in 12 days.
Check The number line shows on which days all three activities will be completed.
Example 2
Find the LCM of 15 and 40.
1. write the prime factorization of each number.
2. Find the common prime factors.
15 = 3 × 5 ← The only common prime factor is 5.
40 = 2 × 2 × 2 × 5
3. Find the product of the prime factors using each common prime factor only once and any remaining factors.
The LCM is 3 × 5 × 2 × 2× 2= 120
Talk Math
Could the LCM of two numbers be one of the numbers? Explain.
Answer:
LCM of two or more numbers may be one of the numbers.
Consider three numbers 3 , 4 , 12
Now, LCM of 3,4 and 12 is 12 which is one of the given numbers.
Guided Practice
Find the LCM of each set of numbers.
Question 1.
6, 10
6: ________
10: _______
So, the LCM of 6 and 10 is _____
Answer:
30
Explanation:
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
10 : 10, 20, 30, 40, 50, 60, 70, 80, 90
So, the LCM of 6 and 10 is 30
Question 2.
3, 4
3: ______
4: _____
So, the LCM of 3 and 4 is ____
Answer:
12
Explanation:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27 and 30.
4: 4, 8, 12, 26, 20, 24, 28, 32, 36 and 40.
So, the LCM of 3 and 4 is 12
Independent Practice
Find the LCM of each set of numbers.
Question 3.
2, 13 ____
Answer:
26
Explanation:
2 : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26
13 : 13, 26, 39, 52, 65, 78, 91, 104, 117
So, the LCM of 2 and 13 is 26
Question 4.
7, 9 ____
Answer:
63
Explanation:
7 : 7 ,14 ,21 ,28 ,35 ,42 ,49 ,56 ,63 ,70
9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90.
So, the LCM of 7 and 9 is 63
Question 5.
2, 10 ____
Answer:
10
Explanation:
2 : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
10 : 10, 20, 30, 40, 50, 60, 70, 80, 90
So, the LCM of 2 and 10 is 10
Question 6.
12, 15 ____
Answer:
60
Explanation:
12 : 12, 24, 36, 48, 60, 72, 84, 96
15 : 15, 30, 45, 60, 75, 90, 105, 120, 135
So, the LCM of 12 and 15 is 60
Question 7.
16, 20 ____
Answer:
80
Explanation:
16 : 16, 32, 48, 64, 80, 96, 112, 128
20 : 20, 40, 60, 80, 100, 120, 140, 160, 180
So, the LCM of 16 and 20 is 80
Question 8.
3, 8 ____
Answer:
24
Explanation:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27
8 : 8, 16, 24, 32, 40, 48, 56, 64
So, the LCM of 3 and 8 is 24
Question 9.
4, 8, 10 ____
Answer:
40
Explanation:
4 : 4, 8, 12, 26, 20, 24, 28, 32, 36 and 40
8 : 8, 16, 24, 32, 40, 48, 56, 64
10 : 10, 20, 30, 40, 50, 60, 70, 80, 90
So, the LCM of 4 , 8 and 10 is 40
Question 10.
3, 9, 18 ____
Answer:
18
Explanation:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27
9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
18 : 18, 36, 54, 72, 90, 108, 126
So, the LCM of 3 , 9 and 18 is 18
Question 11.
15, 25, 75 ____
Answer:
75
Explanation:
15 : 15, 30, 45, 60, 75, 90, 105, 120
25 : 25, 50, 75, 100, 125, 150
75: 75, 150, 225, 300
So, the LCM of 15 , 25 and 75 is 75
Question 12.
9, 12, 15 ___
Answer:
180
Explanation:
9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126, 135, 144, 153, 162, 171, 180
12 :12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180
15 :15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180
So, the LCM of 9 , 12 and 15 is 180
Question 13.
4, 7, 10 ____
Answer:
140
Explanation:
4: 4,8,12,16,20,24,28,32,36,40,44,48,52,56,60,64,68,72,76,80,84,88,92,96,100,104,108,112,116,120,124,128,132,136,140
7: 7,14,21,28,35,42,49,56 ,63,70,77, 84, 91, 98,105,112,119,126,133,140
10 : 10,20,30,40,50,60,70,80,90,100,110,120,130,140,150
So, the LCM of 4, 7and 10 is 140
Question 14.
6, 7, 9 ____
Answer:
126
Explanation:
6 : 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72, 78, 84, 90, 96,102, 108, 114, 120,126
7: 7,14,21,28,35,42,49,56 ,63,70,77, 84, 91, 98,105,112,119,126
9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 108, 117, 126
So, the LCM of 6, 7and 9 is 126
Problem Solving
Question 15.
Juan goes to the bowling alley every 3 weeks. Percy goes to the bowling alley every 5 weeks. If Juan and Percy meet the first time they go bowling, how many weeks will it be before they see each other at the bowling alley again?
Answer:
The first time Percy and Juan meet will be 15 weeks because it is the common multiple of both 5 and 3.
Explanation:
3: 3, 6, 9, 12, 15, 18, 21, 24, 27
5 : 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
So, the LCM of 3 and 5 is 15
Question 16.
A full moon occurs about every 30 days. If the last full moon occurred on a Friday, how many days will pass before a full moon occurs again on a Friday?
Answer:
210
Explanation:
A full moon occurs every 30 days
A Friday occurs every 7 days
You must find out when those days will occur at the same time.
Simplest answer is 7 x 30 = 210 days
7 : 7,14,21,28,35,42,49,56 ,63,70,77, 84, 91, 98,105,112,119,126,133,140,147,154,161,168,175,182,189,196,203,210
30 : 30, 60,90,120,150,180,210
So, the LCM of 7 and 30 is 210
HOT Problems
Question 17.
Mathematical PRACTICE 3 Find the Error Maria is finding the LCM of 6 and 8. Help find and correct her mistake.
Answer:
24
Explanation:
The common prime factors of 6 = 2 × 3
The common prime factors of 8 = 2 × 2 × 2
So only common factor is 2 then,
LCM = 2 × 3 × 2× 2 =24
Question 18.
Mathematical PRACTICE 4 Model Math Write a real-world problem in which it would be helpful to find the least common multiple.
Answer:
An example would be a scenario where you and your friend are going to a a restaurant.
You have lunch there every fourth day, and he has his lunch there every sixth day.
How many days before you meet again for lunch at the same restaurant?
The answer again is lcm(4, 6) = 12.
Question 19.
? Building on the Essential Question How can I find the least common multiple of two or more numbers?
Answer:
The formula to find LCM of two numbers is, LCM of two numbers = product of two numbers ÷ HCF of two numbers or
find all the prime factors of each given number.
List all the prime numbers found, as many times as they occur most often for any one given number.
Multiply the list of prime factors together to find the LCM.
McGraw Hill My Math Grade 5 Chapter 4 Lesson 5 My Homework Answer Key
Practice
Find the LCM of each set of numbers.
Question 1.
7, 14 _____
Answer:
14
Explanation:
7 : 7 ,14 ,21 ,28 ,35 ,42 ,49 ,56 ,63 ,70
14 : 14, 28, 42, 56, 70, 84, 98
So, the LCM of 7 and 14 is 14
Question 2.
6, 15 ____
Answer:
30
Explanation:
6 : 6, 12, 18, 24, 30, 36, 42, 48, 54, 60
15 : 15, 30, 45, 60, 75, 90, 105, 120
So, the LCM of 6 and 15 is 30
Question 3.
5, 9, 15 ____
Answer:
45
Explanation:
5 : 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
9 : 9, 18, 27, 36, 45, 54, 63, 72, 81, 90
15 : 15, 30, 45, 60, 75, 90, 105, 120
So, the LCM of 5 , 9 and 15 is 45
Problem Solving
Question 4.
The cycles for two different events are shown in the table. Each of these events happened in the year 2000. What is the next year in which both will happen?
Answer:
The next year in which both will happen is 2020
Explanation:
4 : 4, 8, 12, 26, 20, 24, 28, 32, 36 and 40
10 : 10, 20, 30, 40, 50, 60
So, the LCM of 4 and 10 is 20
We can add this value to the year when the events occurred together last , that is 2000 and we get 2000 + 20 = 2020
Question 5.
Mathematical Practice 3 Draw a Conclusion Is the statement below always, sometimes, or never true? Give at least two examples to support your reasoning. The LCM of two numbers is the product of the two numbers.
Answer:
Sometimes true
Explanation:
The Least Common Multiple of two numbers is the product of the two numbers only when they are relatively prime.
5, 21 LCM = 105
91, 11 LCM = 1001
If two numbers share a common factor, 8 and 10 share 2 as a common factor,
the least common multiplier of 8 and 10 is 40.
Fill in each blank with the correct word(s) to complete each sentence.
Question 6.
Multiples that are shared by two or more numbers are _____
Answer:
Common multiples
Explanation:
Multiples that are shared by two or more numbers are Common multiples
Question 7.
The least common multiple (LCM) is the ____ multiple, other than 0, common to sets of multiples.
Answer:
Least
Explanation:
The least common multiple (LCM) is the least multiple, other than 0, common to sets of multiples.
Test Practice
Question 8.
Micah is buying items for a birthday party. If he wants to have the same amount of each item, what is the least number of packages of cups he needs to buy?
A. 2 packages
B. 3 packages
C. 4 packages
D. 5 packages
Answer:
C
Explanation:
Multiples of 6 =6, 12, 18, 24, 30, 36, 42, 48, 54, 60
Multiples of 8 =8, 16, 24, 32, 40, 48, 56, 64, 72
LCM of 6 and 8 is 24
Now we will divide 24 by 6 to find number of packages of cups that Micah needs to purchase.
24/6 = 4
Therefore, Micah needs to by 4 package of cups at least.