Deck 11: Number Theory

Question 1

The following sounds all right, but it is not always true. Give a counterexample.
Suppose that k is NOT a factor of m, and k is NOT a factor of n. Then k is NOT a factor of m + n.

Answer 1

There are many possibilities. For example, 3 is not a factor of 4, and 3 is not a factor of 2, but 3 is a factor of 4 + 2 = 6.

Question 2

What, if anything, can you say about the oddness or evenness of m:

A) when 5,063,338 × m is an even number?
B) when 5,063,338 + m is an even number?

Answer 2

A) Nothing can be said about m (the product will be even whether m is even or odd).
B) In this case, m must be even.

Question 3

Five is a multiple of zero. Explain.

Answer 3

False

Question 4

Use the word multiple to say that 360 is a factor of N.

Question 5

If n = 43,759,462,138,999,999,249 + 76,432,157², then is n an even number or an odd number? Explain your answer.

Question 6

Determine if the statements are true or false. Explain your choice.

A) Every whole number is a multiple of itself.
B) It is possible for an even number to have an odd factor.
C) Zero is a multiple of every whole number.
D) 2⁵⁰ is a factor of 100³⁰.

Question 7

Does zero have any factors? Explain your answer.

Question 8

Determine if the following statements are true or false.
A) Every whole number is a factor of itself.
B) It is possible for an odd number to have an even factor.
C) Zero is a factor of every whole number.
D) 5²⁰ is a factor of 50¹².

Question 9

Of what numbers, if any, is zero a multiple? Explain your answer.

Question 10

Suppose 7 is not a factor of n. Can 21 be a factor of n? If 21 can be a factor of n, give an example for n. If 21 cannot be a factor of n, give an explanation from basic principles.

Question 11

Determine whether m and n are primes. Write only enough to make your decisions clear.

A) m = 23 × 29 (= 667)
B) n = 133

Question 12

Is 245 a prime number? Explain.

Question 13

There are no values of b and c for which 27^b = 9^c. Explain your choice.

Question 14

There are no values of r and s for which 11^r = 9^s. Explain your choice.

Question 15

Is there a whole number M that would make this true? If so, tell what M is. If not, tell why not.

A) 3^{5 .} 5^{2 .} 17³ = 3^{4 .} 17^{4 .} M
B) 2^{4 .} 7^{2 .} 11^{8 .} 22 = 2^{5 .} 7 ^. 11^{6 .} M

Question 16

Give the prime factorization of n, where n = 4 × 7²⁰ × 5000. If it is not possible, explain why not.

Question 17

Is it possible to find a nonzero whole number m so that 14^m = 2⁶⁰ × 7⁵⁹? Explain.

Question 18

Note: This is a knowledge of number theory question. Do not use a calculator for this question.
Without calculation, explain why Romeo and Juliet can or cannot both be correct when they are talking about the same large number:
Romeo: "The number is 7 × 11 × 17² × 37 × 67 × 97."
Juliet: "The number is 3 × 11 × 21² × 37 × 67 × 89."

Question 19

Consider this equation: 3,721,164 = 12 × 17² × 29 × 37
Give the prime factorization of 372,116,400 (notice the extra two zeros).
Hint: Do not work too hard.

Question 20

Is the following sentence true? If it is, explain why. If it is not, give a counterexample.
If a number has n factors (n > 1), then the square of the number has 2n factors.

Question 21

When you were a spy, two of your paid informants gave you the following information about the same secret code number:
Informant 1: "The code number is 33 × 70 × some odd number."
Informant 2: "The code number is 35 × 66 × some even number."
What can you tell from your informants' information?

Question 22

Fill in the blanks to make a true sentence or state if no number or algebraic expression will make the sentence true.

A) An example of a number that has an odd number of factors is _____.
B) If n = 13^{8 .} 17¹⁰, then the prime factorization of 26 ^. n is _____.
C) Three will be a factor of 1,400,000,00?,000,000,014 if the missing digit (?) is _____ or _____ or _____.

Question 23

Note: This is a theory question. Do not use a calculator for this question.
Is it possible for some choice of positive whole numbers m and n, such that 45^m = 15ⁿ? Justify your answer.

Question 24

Every two different prime numbers are relatively prime.

Question 25

If we write the first 10,000 numbers in six columns, as started below, then 9999 would be in the fifth column. Write enough (numbers, words) to make your thinking clear.
$\begin{array} { l l l l l l } 1 & 2 & 3 & 4 & 5 & 6 \\7 & 8 & 9 & 10 & 11 & 12\end{array}$

Question 26

Tell the difference between (a) "give a prime factor of 350" versus "give a prime factorization of 350," and (b) "give a number that has an odd factor" versus "give a number that has an odd number of factors."

Question 27

Use the word factor to say that M is a multiple of 240.

Question 28

Put 0 and 2 (one of each) into the blanks to make a true statement. Explain. If it is not possible, explain why.
_____ is a multiple of _____ because _____.

Question 29

If 27 is a factor of n, then n is a multiple of 27.

A) True
B) False
C) It depends on the value of a variable.

Question 30

Give the prime factorization of n, where n = 13⁷ × 3000.

Question 31

In each part, find a whole number for m to make the equality true. If it is not possible, explain why. For credit, your work should show an understanding of number theory.

A) 5² × 10³ × 17⁶ = 2³ × 17⁶ × m
B) 5² × 7⁶ × 11⁴ = 5 × 35⁶ × 11⁴ × m

Question 32

Explain, without extensive calculation, why the following equation can or cannot be correct.
17² × 19² × 37⁵ = 18⁴ × 41⁴
(In your explanation, the grader will look for a clear reference to a major theoretical result.)

Question 33

Note: This is a theory question. Do not use a calculator for this question.
Is it possible, for some choice of positive whole numbers m and n, that 35^m = 25ⁿ? Justify your answer.

Question 34

Name the number of factors of each of these numbers and list them, in factored form.

A) 5² × 17³
B) 3⁵
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Question 35

What is the LARGEST prime number that you need to test in checking for the primeness of the following? Explain your choice.

A) 173
B) 982

Question 36

When the number 540 is written as a product of its prime factors in the form $a ^ { 2 } b ^ { 3 } c$
, what is the numerical value of a + b + c? Choose one of the following:

A) 6
B) 8
C) 10
D) 30

Question 37

Which numbers are prime? If a number is not prime, list at least three factors following the number.

Question 38

Which numbers below divide into 11,220?
2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20

Question 39

A) State a divisibility test for 8.
B) Explain why your test in A will definitely work, using the general seven-digit number, abcdefg, in your explanation.

Question 40

Note: This is a knowledge of number theory question. Do not use a calculator for this question.

A) Select any choice that is a factor of 62,296,715,880 that is equal to
2³ × 3² × 5 × 7 × 47² × 31 × 19².
15 16 21 75 94 217 n = 19 × 31²
B) Explain how you know that your answers in A are correct, even without calculation.

Question 41

Note: This is a knowledge of number theory question. Do not use a calculator for this question.
Select any choice that is a factor of 80,000,000,005,332.
3 4 5 6 8 9 12 15

Question 42

For each part, give an example, if one exists. If there is no example, explain why not.

A) A whole number that has 15, 21, and 1000 as factors but does not have 9 as a factor
B) A prime number that has 7 and 19 as factors and is not a perfect square

Question 43

Note: This is a knowledge of number theory question. Do not use a calculator for this question.
Select each of the given choices that is a factor of the given number n.
n = 2^{2 .} 10^{3 .} 7^{11 .} 13⁵
$\text { Choices: } 8 \quad 14 \quad 21 \quad 28 \quad 35$

Question 44

If it is possible, give a whole number that is relatively prime to 24. If it is not possible, explain why.

Question 45

State a divisibility test for 4, and explain why it works.

Question 46

Note: This is a knowledge of number theory question. Do not use a calculator for this question.
Is it possible for some choice of positive whole numbers m and n that 75^m = 25ⁿ? Justify your decision.

Question 47

Write the prime factorization of the following. (Show your work.)

A) 1485
B) 792
C) Name all common factors of 1485 and 792. (They can be in factored form.)
D) What is the greatest common prime factor of 1485 and 792?

Question 48

A) What is the least common multiple of 1485 and 792 (in factored form)?
B) Write two other common multiples of 1485 and 792.

Question 49

Suppose K = $2 ^ { 5 } \cdot 7 \cdot 11$ , L = $2 ^ { 3 } \cdot 7 \cdot 11 \cdot 13$ , M = $2 \cdot 29 ^ { 2 }$ , and N = $4 \cdot 11 \cdot 13 ^ { 2 } \cdot 29$ .
Name the least common multiple of each of the following (in factored form).

A) K and L
B) M and N
C) K and M
D) K, L, and N

Question 50

Suppose K = $2 ^ { 5 } \cdot 7 \cdot 11$ , L = $2 ^ { 3 } \cdot 7 \cdot 11 \cdot 13$ , M = $2 \cdot 29 ^ { 2 }$ , and N = $4 \cdot 11 \cdot 13 ^ { 2 } \cdot 29$
. Name the greatest common factor of each of the following (in factored form).

A) K and N
B) K and L
C) M and N
D) K, L, and M

Question 51

Write these numbers in simplest form.

A) $\frac { 26 } { 65 }$

B) $\frac { 616 } { 792 }$

C) $\frac { 129 } { 215 }$

Question 52

A) Use the prime factorizations of 345, 264, and 495 to find the least common multiple of the three numbers.
B) Compute the following: $\frac { 345 } { 495 } + \frac { 250 } { 264 }$ . (Leave the answer in factored form.)

Question 53

Two neighboring satellites send out signals at regular intervals. One sends a signal every 180 seconds, and the other sends a signal every 280 seconds. If both satellites send out a signal at 12:00 midnight on January 1, when will be the next time that they both send out a signal at the same time?

Question 54

Hamburger patties come in packages of 16, and hamburger buns come in bags of 12. How many of each do you need to buy so that you have the same number of buns as you do hamburgers?

Question 55

As a charitable service, your class undertakes a project where they fill backpacks with donated school supplies for underprivileged children. The donations include 135 notebooks, 216 pencils, and 81 pens. You want to use all the donations and include the same number of each item in each backpack. What is the largest number of backpacks you can fill, and how many items will be in each backpack?

Question 56

Two football players are working out by running around a track. The first can run the track in 3 minutes, and the second one can run the track in 4 minutes. If they begin at the starting point at the same time and run in the same direction at the same rates, when will they both be at the starting point again?

Question 57

A band has been invited to march at the Rose Parade and needs to make money to cover the expenses. They divide up into three teams and shovel snow from long driveways for four days before Christmas. The first team makes $315, the second $240, and the third $210. If they charged the same whole-dollar rate for each driveway, what was that rate? Explain.