The document contains a mathematics program focused on grade 5 concepts, including multiplication, division, prime numbers, and factorization. It provides examples, practice problems, and word problems that require finding sums, averages, least common multiples, and greatest common factors. It also emphasizes the concept of prime factorization and its applications in various mathematical scenarios.

1. MTAP Program of Excellence in
Mathematics
Grade 5 Session 2
Maria Coritha A. Montemayor

2. MULTIPLICATION AND DIVISION
OF WHOLE NUMBERS

3. A. Multiply the following.
732
34
x
135
24
x
626
72
x
474
43
x
856
17
x
8
2
9
1
2
6
9
1
2
8
8
8
4
2
1
0
4
5
0
7
2
0
4
2
3
2
5
2
1
2
8
3
4
2
7
0
5
4
2
2
4
1
6
9
8
1
2
8
3
0
2
2
9
9
5
6
5
8
2
5
5
4
1
2
1
1
1
4
1. 2. 3. 4. 5.
1
1
1
1
1
2
1
2
1
4
3
1
1

4. B. Divide the following.
5
45
−
09
1
9
−
0
2
34
−
51
3
5 1
−
0
38
−
108
95
−
133
133
−
0
257
43
−
227
215
−
129
129
−
0
153
795
−
1113
1113
−
0
57

5. C. Find the quotients mentally.
812
5450
651
764
397
147
247
3698
1562
4056

6. .
PRIME NUMBERS
• In other words, the prime number is a positive
integer greater than 1 that has exactly two factors,
1 and the number itself.
• There are many prime numbers, such as 2, 3, 5, 7,
11, 13, etc.
• Keep in mind that 1 cannot be either prime or
composite.
• The remaining numbers, except for 1, are
classified as prime and composite numbers.

7. .
Example: What are the prime factors of 12 ?
It is best to start working from the smallest prime number, which is 2, so let's check:
12 ÷ 2 = 6
Yes, it divided exactly by 2. We have taken the first step!
But 6 is not a prime number, so we need to go further. Let's try 2 again:
6 ÷ 2 = 3
Yes, that worked also. And 3 is a prime number, so we have the answer:
12 = 2 × 2 × 3
As you can see, every factor is a prime number, so the answer must be right.
Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 =22 × 3

8. _______1. 220 _______ 5. 121 ______ 9. 101 _______13. 702
_______2. 83 _______6. 107 _______10. 301 _______14. 211
_______3. 105 _______7. 109 _______11. 723 ______ 15. 311
_______4. 223 _______8. 117 _______12. 517 _______16. 501
C. Which of the following are prime numbers? Write P
on the blank provided before the number.
𝐏
𝐏
𝐏
𝐏
𝐏
𝐏
𝐏
𝑷𝒓𝒊𝒎𝒆 𝑵𝒖𝒎𝒃𝒆𝒓𝒔 −a whole number greater than 1 that cannot be exactly
divided by any whole number other than itself and 1 (e.g. 2, 3, 5, 7, 11).

9. .
PRIME FACTORIZATION
• is breaking a number down into the set of prime
numbers which multiply together to result in the
original number. This is also known as prime
decomposition. We cover two methods of prime
factorization: find primes by trial division, and use
primes to create a prime factors tree.

10. .
Example: What is the prime factorization of 147 ? Can we divide 147 exactly by 2?
147 ÷ 2 = 73½ No it can't. The answer should be a whole number, and 73½ is not.
Let's try the next prime number, 3: 147 ÷ 3 = 49
That worked, now we try factoring 49.
The next prime, 5, does not work. But 7 does, so we get: 49 ÷ 7 = 7
And that is as far as we need to go, because all the factors are prime numbers.
147 = 3 × 7 × 7 (or 147 = 3 × 72 using exponents)
Example: What is the prime factorization of 17 ?
Hang on ... 17 is a Prime Number. So that is as far as we can go. 17 = 17

11. .

12. D. Write the prime factorization of each of the following
numbers.
1. 252 5. 360 9. 2240
2. 104 6. 624 10. 1536
3. 264 7. 8072 11. 2592
4. 450 8. 1728 12. 1512
= 22 x 32 x 7
= 23 x 13
= 23 x 3 x 11
= 2 x 32 x 52
= 23 x 32 x 5
= 24 x 3 x13
= 23 x 1009
= 26 x 33
= 26 x 5 x 7
= 29 x 3
= 25 x 34
= 23 x 33 x 7

13. SUM AND AVERAGE
.

14. E. Find the sum of set of number and give the average.
Explain how to get the average of a given set of numbers.
SET OF NUMBERS SUM AVERAGE
1. 24, 18 , 33
2. 36 , 54 , 83 , 27
3. 34 , 42 , 47
4. 82 , 47 , 55 , 92
6. 248 , 123 , 217 , 244
(24 + 18 + 33) = 75
(36 + 54 + 83 + 27) = 200
(34 + 42 + 47) = 123
(82 + 47 + 55 + 92) = 276
(248 + 123 + 217 + 244)= 832
(24 + 18 + 33)/3 = 75/3 = 25
(36 + 54 + 83 + 27)/4 = 200/4 = 50
(34 + 42 + 47)/3 = 123/3 = 41
(82 + 47 + 55 + 92)/4 = 276/4 = 69
(248 + 123 + 217 + 244)/4 = 832/4 = 208

19. E. Give the GCF and LCM of each pair of numbers.
Given Number GCF LCM
1. 42 , 72
2. 44 , 55
3. 154 , 264
4. 88 , 96
5. 40 , 60
6. 100 , 280
7. 120 , 180
8. 144 , 216
9. 120 , 150
6
11
22
8
20
20
60
72
30
504
220
1848
1056
120
1400
360
432
600

20. SOLVING WORD
PROBLEMS

21. F. Answer each of the following problems:
1. At the gym, Rose swims every 6 days, runs every 4 days, and cycles
every 16 days. If she did all three activities today, in how many days will she
do all three activities again on the same day?
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52.
6: 6, 12, 18, 24, 30, 36, 42, 48, 54.
16: 16, 32, 48.
In 48 days, she will do all three on the same day.
Answer:

22. 2. Sean needs to ship 14 rock CDs, 12 classical CDs, and 8 pop CDs. He can
pack only one type of CD in each box, and he must pack the same number of
CDs in each box. What is the greatest number of CDs Oscar can pack in each
box?
Factors of
14: 1, 2, 7, 14
12: 1, 2, 3, 4, 6, 12
8: 1, 2, 4, 8
Sean can pack 2 CDs in each box.
Answer:

23. 3. I want to plant 45 daisy plants, 81 mango plants and 63 eggplant plants in my
garden. If I put the same number of plants in each row and each row has only
one type of plant, what is the greatest number of plants I can put in one row?
Factors of
45: 1, 3, 5, 9, 15, 45
81: 1, 3, 9, 27, 81
63: 1, 3, 7, 9, 21, 63
I can put 9 in each row.
Answer:

24. 4. Forks are sold 6 to a package and Spoons are sold 8 to a package. If you
want to have the same number of each item for a party, what is the least
number of packages of each you need to buy?
Multiples of 6:
6, 12, 18, 24, 30, 36, 42, 48, 54.
8: 8, 16, 24, 32, 40, 48, 56, 64, 72
He needs to buy at least 24 of each, so 4 packages forks
and 3 packages of spoons. He needs to buy 4 packages of forks
and 3 packages of spoons.
Answer:

25. 5. Zoie collects stickers. She can arrange those 15 or 18 stickers to a page
without any left over. What is the least number of stickers for which she can
do this?
Answer:
Least number of stickers for which she can do this is equal to the LCM
of 15 and 18 which is 90.

26. 6. Every composite number can be written as a product of its prime factors.
What is the smallest number that has 2, 3 and 5 as factors?
7. What is the smallest number that has 2, 3 and 7 as factors? How many
numbers less than 300 have 2, 3 and 7 as factors?
The smallest number that has 2, 3 and 5 as factors is 2 x 3 x 5 = 30.
Answer:
Answer:
The smallest number that has 2, 3, and 7 as factors is 2 x 3 x 7 = 42.
The numbers less than 200 with 2, 3 and 7 as factors are 42, 84, 126, 168,
210, 252, and 294.

27. 8. Every even number greater than 2 can be written as the sum of 2 or more
prime numbers. For examples, 6 = 3+3, 8 = 3 + 5. Write 12, 16, 22, 24 and 34
as the sum of two or more prime numbers.
12 = 5 + 7;
16 = 5 + 11;
22 = 5 + 17 or 3 + 19;
24 = 7 + 17 or 5 + 19;
34 = 3 + 31 or 5 + 29
Answer:

28. 9. A full moon occurs 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:
To find a common multiple, we can multiply 30 days by 7 days in a
Week (to end up on Friday again).
30 x 7 = 210 It will take 210 days.

29. 10. Jhamia has diving lessons every fifth day and swimming lessons every third
day. If she had a diving lesson and a swimming lesson on May 5, when will be
the next date on which she has both diving and swimming lessons?
5 x 3 = 15 LCM = 15 5 + 15 = 20
Jhamia will have both diving and swimming lessons on May 20th
Answer:

30. CHALLENGE!

31. Let us apply what we learned!

32. Solve the following problems.
1. Red can bike around their subdivision in 4 minutes while
Jim can do it is 5 minutes. If they start at the gate of their
subdivision at 3:00 P.M., when will they be together again at
the gate? How many times will they be together at the gate
in one hour?
Red will be at the gate at 3:04, 3:08, 3:12, 3:16, 3:20, etc. while Jim will be
at the gate at 3:05, 3:10, 3:15, 3:20, 3:25, etc. They will be at the gate together at
3:20. They will be together at gate 3 times in one hour; at 3:20, 3:40 and 4:00. In one
hour, they will be at the gate three times.

33. 2. There are red, blue and green bulbs in a parole. The red
blinks every 2 seconds, the blue blinks every 3 seconds
and the green blinks every 4 seconds. They first blink
together at 8:00 P.M. When will they blink together again?
How many times will they blink together in one minute?
The red will blink at 8:00, 8:02, 8:04, 8:06, 8:08, 8:10, 8:12, 8:14, 8:16, 8:18, 8:20, 8:22,
8:24, 8:26, etc.
The blue will blink at 8:00, 8:03, 8:06, 8:09, 8:12, 8:15, 8:18, 8:21, 8:24, 8:27, 8:30, etc.
The green will blink at 8:00, 8:04, 8:08, 8:12, 8:16, 8:20; 8:24, 8:28, 8:32, etc.
The list shows they will blink together again at 8:12, 8:24, etc. They blink together 5
times an hour.

34. 3. In a test, Jane scored 15 points fewer than Susan who
scored 78 points. Lianne scored two-thirds as many points
as Jane. How many points did Jane and Lianne score?
Susan = 78 points ,
Jane scored 78 – 15 = 63 points, and
Lianne scored 63 x
2
3
= 42 points

35. 4. J = 1 + 12 + 13 + 14 + 15 + 16 + 17 + 18. How many
prime factors does J have?
J = 1 + 12 + 13 + 14 + 15 + 16 + 17 + 18 = 8.
8 = 2 x 2 x 2, therefore it has 1 prime factor. (2)

36. 5. A clock alarms at the indicated number of times at regular
intervals as follows: ..., 8, 1, 9, 1, 10, 1, 11, 1, 12, 1, 1, 1, _____,
______, … What are the next two numbers?
The answer is 2 and 1.