This document provides information about determining divisibility rules for numbers 1-11 and examples for each. It then has students check which numbers on a list are divisible by each number 1-11. Next, it categorizes numbers as prime, composite, odd or even. Students are asked to find the prime factorization of various numbers using exponents. They also find the greatest common factor and least common multiple of pairs of numbers. Word problems involving these concepts are presented along with challenge problems about biking times, twin primes, and properties of greatest common factors and least common multiples.

1. 2023MTAP
Saturday Program in Mathematics
GRADE 4 SESSION 4

2. A. Describe how you will determine when a number is
divisible by 1. 2? 2. 3? 3. 4? 4. 5? 5. 6? 6. 7? 7. 8? 8. 9?
9. 10? 10. 11? Give an example of each.

3. 1.
2.
3.
4.
5.
A number is divisible by 2 if it is even or it ends in 2, 4, 6, 8, 0, i. e,
56, 78, 98, 80, etc.
A number is divisible by 3 if the sum of the digits of the number is divisible by
3, e.g. 324/3 = 108 and 3 + 2 + 4 = 9 and 9 is divisible by 3.
A number is divisible by 4 if the number formed by the last 2 digits is divisible by
4, e.g. 124, 388, 3 248, 5248, or the number ends in 2 zeros.
A number is divisible by 5 if it ends in 0 or 5; e.g. 60, 5485, 2390
A number is divisible y 6, if it is divisible by both 2 and 3; e.g. 492, 876, 1272.

4. 6.
7.
8.
9.
10.
A number is divisible by 7: Subtract twice the last digit from the rest of the digits; repeat
as necessary until the last difference is a 1-digit or 2-dgit number that is divisible by 7.
Take 69125, 6912 – 10 = 6902, 690 – 4 = 686, 68 – 12 = 56. Since 56 is divisible by 7, 69125
is divisible by 7
A number is divisible by 8 if the number formed by the last 3 digits is divisible by 8, or
the number ends in three zeros. In 80 656, 656 ÷ 8 = 82, so 80 656 is divisible by 8.
A number is divisible by 9 if the sum of the digits is divisible by 9. 311 040 is
divisible by 9 since 3+1+1+4 = 9. 311 040 ÷ 9 = 34560
A number is divisible by 10 if it ends in 0. 56 780 ÷ 10 = 5 678.
A number is divisible by 11 if the sum of the digits in the odd places and the
sum of the digits in the even places are equal or they differ by a multiple of 11.
In 87 648, 8 + 6 + 8 = 22, 7 + 4 = 11. 22 – 11 = 11, so, 87 648 is divisible by 11

5. B. Which of the following are divisible by 2? 3? 4? 5? 6? 7? 8? 9? 10? 11? Check the
corresponding column.
1. 55
2. 33
3. 48
4. 840
5. 420
6. 610
7. 252
8. 1 980
9. 2 780
10. 1 125
2 3 4 5 6 7 8 9 10 11
✔ ✔
✔ ✔
✔ ✔ ✔ ✔ ✔
✔ ✔ ✔ ✔ ✔ ✔ ✔
✔ ✔ ✔ ✔ ✔ ✔ ✔
✔ ✔ ✔
✔ ✔ ✔ ✔ ✔ ✔
✔ ✔ ✔ ✔ ✔ ✔ ✔
✔ ✔ ✔ ✔
✔ ✔ ✔

6. C. Which of the following are prime numbers, composite numbers, odd
numbers or even numbers? List them under the proper group to which
they belong in your notebook. Note that a number like 2 may belong to
more than one group.
11 20 3 10 59 100 85 53 52 39
71 45 89 61 18 69 73 35 37 44
29 48 90 43 57 9 79 69 49 91

7. PRIME NUMBERS
COMPOSITE
NUMBERS ODD NUMBERS EVEN NUMBERS
59, 11, 3,
53, 71, 89,
61, 73, 37,
29, 43
20, 10, 100,
85, 45, 39, 52,
18, 63, 35, 44,
48, 90, 57, 9,
69, 49, 91
3, 11, 59, 85,
53, 39, 71, 45,
89, 61, 69, 73,
35, 37, 29, 43,
57, 9, 69, 49,
91
20, 10, 100,
52, 18, 44, 48,
90

8. D. Give the prime factorization of the following numbers using exponents
where appropriate.
1. 88 =
2. 44 =
3. 68 =
4. 98 =
5. 58 =
6. 384 =
7. 630 =
8. 96 =
9. 153 =
10. 350 =
2³ x 11
2² x 11
2² x 17
2 x 7²
2 x 29
2⁷ x 3
2 x 3² x 5 x 7
2⁵ x 3
3² x 17
2 x 5² x 7

9. D. Give the prime factorization of the following numbers using exponents
where appropriate.
11. 816 =
12. 108 =
13. 858 =
14. 1800 =
15. 448 =
16. 180 =
17. 1080 =
18. 1050 =
19. 2520 =
20. 1968 =
2⁴ x 3 x 17
2² x 3³
2 x 3 x 143
2³ x 3² x 5²
2⁶ x 7
2² x 3² x 5
2³ x 3³ x 5
2 x 3 x 5² x 7
2³ x 3² x 5 x 7
2⁴ x 3 x 41

10. E. Give the GCF of the following pairs of numbers.
1. 24 and 36 -
2. 34 and 51 -
3. 12 and 18 -
4. 84 and 96 -
5. 16 and 40 -
6. 54 and 78 -
7. 96 and 132 -
8. 84 and 112 -
12
17
6
12
8
6
12
28

11. F. Give the LCM of the following pairs of numbers.
1. 40 and 64 -
2. 36 and 40 -
3. 36 and 81 -
4. 27 and 81 -
5. 16 and 40 -
6. 60 and 72 -
7. 84 and 105 -
8. 48 and 69 -
320
360
324
81
80
360
420
240

12. G. Answer each of the following problems.
1. There are a total of 554 scouts in a school composed of Kab Scouts, Boy Scouts and Girl
Scouts. If there are 234 Boy Scouts, 212 Girl Scouts, how many Kab Scouts are there?
2. Grace and Flor collected 457 popsicle sticks for their project. If Grace collected 269 popsicle
sticks, how many did Flor collect?
3. What are the prime factors of 105?
4. Find a two-digit composite number that is the difference of the squares of two prime numbers?
5. What is the biggest factor that any number can have?
108 Kab Scouts
188 popsicle sticks.
3, 5, 7
192 – 172 = 361 – 269 = 92; 132 – 112 = 169 – 121 = 48
The biggest factor any number can have is itself.

13. Challenge:
1. Tom and John can bike around a subdivision in 8 minutes and 6 minutes respectively. If they
start together at 8:00 one morning, what time will they be together again at the starting point?
2. Twin primes, like 3 and 5, are two prime numbers that differ by 2. Find three sets of twin primes.
3. a. Consider 24 and 32.
a. Express 24 and 32 as products of prime factors.
b. Find the GCF and LCM of 24 and 32.
c. Find 24 x 32 and divide the product by their GCF.
d. Repeat a – c using 18 and 28.
together again at the starting point at 24 and 48 minutes after 8:00 a.m
2 and 3, 3 and 5, 5 and 7, 11 and 13, 17 and 19 and so on
24 = 2 x 2 x 2 x 3 ; 32 = 2 x 2 x 2 x 2 x 2
GCF (24, 32) = 8, LCM = 2% x 3 = 96
768; 768/8 = 96 which is the LCM of 24 and 32.
18 = 2 x 3 x3, 28 = 2 x 2 x 7 GCF(18 & 28) = 2 LCM( 18, 28) = 18 x 2 x 7 = 252
18 x 28 = 504; 504/2 = 252 which is the LCM of 18 and 28
e. What do you find? Will what you found in d true for any two composite numbers?
This result is always true.

14. Challenge:
4. The GCF of two numbers is 8 and their LCM is 384. If one number is 16, what is the other
number?
5. If the product of two prime numbers is 437, what are the two numbers?
GCF is 8, the LCM = 384 and one number is 16. From 3 e, GCF x LCM = ab.
11 and 37

15. - END -