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Prime Time
Looking Back Answers
1. a. Yes; during the 30th week of use.
b. Yes; during the 24th week of use.
c. Brandon is correct. Since the secret
number is a factor of 90, and 90 is a
factor of 180, then the secret number
is a factor of 180.
d. Clue 2 is still not enough to
determine the number since 90 has
three prime factors: 2, 3, and 5.
e. Yes; since 21 is a multiple of 3 but
not a multiple of 2 or 5, the secret
number is 3.
2. a. There are six factor pairs for 60, but
since 1 × 60 and 60 × 1 are different
arrangements for a spectator
looking at the band, there are twelve
arrangements. The twelve rectangles
have dimensions:
1 × 60; 2 × 30; 3 × 20; 4 × 15;
5 × 12; 6 × 10;
10 × 6; 12 × 5; 15 × 4; 20 × 3;
30 × 2; 60 × 1
5. 3 × (2 + 1) = 9
6. (6 + 4 + 3) × 2 = 26
b. Since 61 is prime, the only two
arrangements possible are 1 × 61
and 61 × 1.
7. (5 × 7) – (2 × 7) = 21
3. a. The LCM is 9,900.
b. The GCF is 3.
c. The factors of Tamika’s special
number are 1, 2, 3, 4, 6, 11, 12, 22, 33,
44, 66, and 132.
d. A number is even if its prime
factorization contains the number 2.
If 2 is not in the prime factorization,
then the number is odd. So, Tamika’s
number is even, and Cyrah’s number
is odd.
e. A number is a square number if each
number in its prime factorization
occurs an even number of times.
Cyrah’s number is a square number
since the prime factors 3 and 5 occur
twice each. Tamika’s number is not a
square number.
4. a. No, this clue is not enough since 90
has several factors.
8. a. (1) Find all of the factor pairs
(stopping when the pairs start
to repeat). (2) Use the prime
factorization of the number.
(3) To start a list of factors, use clues
such as: Is the number even? Odd?
A multiple of 5 or 10? (4) Make
rectangles whose dimensions are the
factor pairs of the number.
b. (1) Make a list of the multiples of
each number starting with the
number itself. The least common
multiple is the first multiple to
appear in each list. For example, in
Question 3, the multiples of Tamika’s
number are 132; 264; 396; 528; 660;
792; 924; 1,056; 1,188; 1,320; 1,452;
1,584; 1,716; 1,848; 1,980; 2,112; . . . ;
9,768; 9,900; 10,032; . . . .
The multiples of Cyrah’s number
are 225; 450; 675; 900; 1,125; 1,350;
1,575; 1,800; 2,025; . . . ; 9,675; 9,900;
10,125; . . . .
b. The least possible secret number
is 1. The greatest possible secret
number is 90.
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Prime Time
Looking Back Answers (continued)
(2) Use the prime factorization.
The LCM must contain each of
the primes that occur in either
number. The prime factor must
occur the maximum number
of times it occurred in either
number. For example, in
Question 4, the LCM of Tamika’s
number and Cyrah’s number is
2 × 2 × 3 × 3 × 5 × 5 × 11, or 9,900.
Note: Discussing both strategies
may reveal that the prime
factorization may be the most
efficient method for finding the
LCM. Also refer to Question 2,
which is an application that involves
finding the LCM of two numbers.
c. (1) List all of the factors of each
number and then find the greatest
factor that is common to both lists.
For example, in Question 3, the
factors of Tamika’s number are 1,
2, 3, 4, 6, 11, 12, 22, 33, 44, 66, and
132. The factors of Cyrah’s number
are 1, 3, 5, 9, 15, 25, 45, 75, and 225.
(2) Use the prime factorization. The
GCF must contain all of the prime
factors that occur in each number.
In Question 3, the GCF is 3. It is
the only prime that occurs in both
factorizations.
9. a. A number is prime if its only factors
are 1 and itself. One way to decide
whether or not a number is prime is
to list all the factors of the number.
b. (1) A number is square if its prime
factorization contains each prime
factor an even number of times. See
Question 3 part (e) for an example.
(2) Some students will find all the
factors and decide that if the number
has an odd number of factors, then it
is a square number.
(3) Some students may guess and
check to see if the number has a
factor pair that contains two equal
factors. In this case, they need to
check only the numbers 2 through
the whole number that is the closest
to the square root of the number.
(4) Some students may suggest
using the square root button on a
calculator.
(5) Some students may say that a
number is square if you can make a
square whose area (number of unit
tiles) is the number.
c. A number is even if 2 is a factor of
the number.
d. A number is odd if 2 is not a factor of
the number.
10. a. False. This is not true in general. For
example, 12 is greater than 10, and
12 has six factors, 1, 2, 3, 4, 6, and 12,
while 10 has only four factors: 1, 2, 5,
and 10. However, 29 is greater than
6, but 29 has only two factors, 29 and
1, while 6 has four factors: 1, 2, 3,
and 6.
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Prime Time
Looking Back Answers (continued)
b. True. The sum of two odd numbers
is always even. An odd number can
be split into groups of two with one
leftover. Adding two odd numbers
will result in two “leftovers,” or
another group of 2. An illustration is
a good way to demonstrate this.
Note: If X and Y are odd, then
X = 2N + 1 and Y = 2M + 1, where
N and M are whole numbers. Thus,
X + Y = 2N + 1 + 2M + 1
= 2N + 2M + 2
= 2(N + M + 1)
So, the sum is even.
c. False. Students may use a geometric
argument using rectangular
arrangements. For example,
4 × 3 = 12. You have four rows of
three. Thus, the product contains
four groups of three, which make an
even number of threes in the product.
Note: If X is even, then X = 2N. If
Y is odd, then Y = 2M + 1. Then
XY = 2N(2M + 1). This shows that
2 is factor of XY; thus, XY is even.
d. True. For example, if the two
numbers are 5 and 17, then the LCM
must contain all of the common
factors of 5 and 17, but since both
are prime, the LCM is 5 × 17.
Note: The above reasoning is similar
to the following general argument: If
x and y are prime, x is the only factor
of x other than 1, and y is the only
factor of y other than 1. Therefore,
the first common multiple with both
x and y as factors is xy.
e. False. This is not true in general.
If x = 12 and y = 18, then the GCF
is 6, and 6 is less than both 12 and
18. On the other hand, if one of the
numbers is a factor of the other, that
number is the greatest common
factor of the two. For example, if
x = 6 and y = 12, then the GCF of
6 and 12 is 6.
11. In arithmetic calculations with multiple
operations, performing the operations
in different orders sometimes leads
to different answers. The Order of
Operations convention is useful in
deciding which operation should be
performed first. Using the Distributive
Property allows arithmetic calculations
to be done in two different ways, with
equivalent expressions in factored
form and expanded form, resulting
in the same answer. These properties
of operations on numbers are useful
in helping people perform arithmetic
calculations correctly while still
allowing for multiple ways to arrive
at a correct answer.
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