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GCF LCM problems


For all the GCF/ LCM problems , you are expected to use (and to show) the methods we learned in class.
Assume all letters represent counting numbers.

1. Indicate whether each statement is true or false:
a) 6 is a multiple of 12.
b) Any multiple of 9 will also be a multiple of 12.
c) If a number is made up of 18’s, it is also made up of 4’s.
d) Any counting number that is not prime is composite.
e) 93,198,624 is not divisible by 6.
f) 1 and m are relatively prime.
g) If two different numbers are prime, then they must be relatively prime.
h) Any counting number which is not prime must be composite.
i) The GCF of 6x and 7x is x.
j) Thc LCM of two different prime numbers is their product.
k) The LCM of n and 1 is 1.

a) If LCM(x, y) = 16, then x is a multiple of 16.
b) If a and b are relatively prime, then GCF(a, b) = b.
c) If GCF(m, n) = 6, then n is a factor of 6.
d) If a is a factor of b and
a ≠ b, LCM(a, b) = b.
e) GCF(a, 1) = a.
f) If a and b are both multiples of 3, they are relatively prime.
g) Two consecutive odd numbers are always relatively prime.
h) The product of two different prime numbers is always a prime number.
i) There is a largest prime number.
j) If n is a multiple of m, then some, but not all, multiples of n are also multiples of m.

2. Suppose you can chop a number, x, into 11 different parts, and show that each of the 11 parts is
made of 7’s (with nothing left). Then what can you conclude about the number x?

3. Is 37,871,700 divisible by 4? Without doing long division, and without using the divisibility rule
for 4, how can you tell?

4. For the number
(where p is a prime number greater than 7), indicate whether the
following statements are true or false:

a) m is odd
b) the last digit (on the right) of the number m is a 5
c) 14|m
d) 4|m
e) m is a multiple of 490
f) m is divisible by 7p


1. List all the factors of 48.
2. Write the prime factorization of 630.
2. Systematically list all the factors of . Write them in their prime factorizations.
3. Systematically list all the factors of . Write them in their prime factorizations.
4. How many factors does have? (Show work.) Write any 4 of them, in their prime
factorizations.

5. Name two numbers, both above 20, that are relatively prime.
6. Explain: any multiple of 5 that is greater than 5 must be composite.
7. What is the definition of composite?
8. Find the smallest composite number which is not divisible by any counting number between 1 and
50.
9. Name two different numbers, both between 20 and 300, that have a GCF of 15.
10. Name two different numbers that have an LCM of 30.
11. Find, using the technique we used in class, LCM(24, 45).
12. Find, using the technique we used in class, GCF(350, 630).

13. In the problem 6x4=24 , 6 is called the _________, 4 is called the _______, and 24 is called the
_________.
14. In the problem 6+4=10 , 6 is called the _________, 4 is called the _______, and 10 is called the
_________.
15. In the problem 6-4=2 , 6 is called the _________, 4 is called the _______, and 2 is called the
_________.
16. In the problem , 12 is called the _________, 4 is called the _______, and 3 is called the
_________.
17. Name four pairs of numbers that have an LCM of 100.

18. What is the name of the “filter” which can be used to find prime numbers in the first n counting
numbers? Briefly explain how one uses this “filter.”

19. What pairs of numbers have a GCF of 5 and an LCM of 125? (List all of the possibilities.)

20. If GCF (x, 72) = 24 and LCM (x, 72) = 360, find x.
21. If GCF (x, 60) = 12 and LCM (x, 60) = 480, find x.
22. If GCF (x, 36) = 12 and LCM (x, 36) = 252, find x.
23. If GCF(x, 56) = 8 and LCM(x, 56) = 504, find x.
24. If GCF(m, 52)= 4 and LCM(m, 52)= 520, find m.
25. Name two numbers between 200 and 1000 that have a GCF of 12.

26. Evaluate:
a) GCF (x, 3x) =
b) LCM (b, 4b) =
c) GCF (a, 4 ) =
d) LCM (1, n) =
e) GCF(5a, 4a) =
f) LCM(ab, 1) =
g) LCM(1, a, 3a) =

27. If the GCF of two numbers is 8 and their LCM is 320, what could the numbers be? (List all the
possibilities.)
28. Name three pairs of numbers x and y, so that neither x nor y is a multiple of 3, and GCF(x, y) = 4.
(Your x’s and y’s should be 6 different numbers.)
29. Name three pairs of different odd numbers that have a GCF of 15. (Your x’s and y’s should be 6
different numbers.)
30. Find two numbers that have a GCF of 30 and an LCM of 450. List all the possibilities.

31. Find two numbers m and n so that m and n are both greater than 100, , and GCF(m, n) = 33.
32. Find 3 different pairs of numbers x and y (6 different numbers), so that LCM(x, y) = 200.
33. Find all possible pairs of numbers that have an LCM of 121.
34. Find a pair of numbers that has a GCF of 12 and an LCM of 420.
35. Use the rules for divisibility to see if 487,342 is divisible by 3, 4, 6, 8, and 9. (Be sure to show that
you know what each rule is and that you show how you are using each rule.)

36. Briefly, but clearly, explain why the divisibility rule for 8 works.
37. Briefly, but clearly, explain why the divisibility rule for 3 works.
38. Name 4 natural objects that contain Fibonacci numbers.
39. Name the first 20 Fibonacci numbers.
40. Name any 3 mathematical facts (from the posted list) about Fibonacci numbers.

41. Write the first 10 rows of Pascal’s Triangle.
42. Briefly describe 6 characteristics of or patterns in Pascal’s Triangle.
43. Find the probability of getting at least 6 heads when tossing a coin 8 times .
44. Briefly describe the “Split Screen” activity. Comment on the grade levels for which this activity is
appropriate.
45. Briefly describe the “Buzz Bop Bing” activity. Comment on the grade levels for which this activity
is appropriate.

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