Which numbers are prime? If a number is not prime, list at least three factors following the number. $\begin{array} { l l l l } 39 ^ { 2 } & 5,231,211 & 61 \cdot 73 & 121&43 \end{array}$

The answer of Which numbers are prime? If a number...

Exam 11: Number Theory

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?

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Question 1

Correct Answer:

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Their information is inconsistent. Although the given factors (33 × 70 and 35 × 66) do give the same prime factorizations, informant 2's "even number" would involve another factor of 2 that informant 1's odd number could not.

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.

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Question 2

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No, this is not true. Students should have no trouble finding a counterexample. (The question is just testing the ability to read an if-then statement and of knowing what a counterexample is.)

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

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Question 3

Correct Answer:

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No. Since 14 = 2 × 7, 14 to any power will have the same number of twos as sevens.

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.

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Question 4

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.

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Question 5

Write these numbers in simplest form. A) $\frac { 26 } { 65 }$ B) $\frac { 616 } { 792 }$ C) $\frac { 129 } { 215 }$

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Question 6

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

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Question 7

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 .} 10³^. 7^{11 .} 13⁵ $\text { Choices: } 8 \quad 14 \quad 21 \quad 28 \quad 35$

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Question 8

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

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Question 9

Which numbers are prime? If a number is not prime, list at least three factors following the number. 3 5,231,211 61\cdot73 121 43

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Question 10

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

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Question 11

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?

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Question 12

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.

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Question 13

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.

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Question 14

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 15

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:

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Question 16

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

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Question 17

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?

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Question 18

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

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Question 19

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.

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Question 20

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Exam 11: Number Theory

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?

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.

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

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 35m = 25n? Justify your answer.

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.

Write these numbers in simplest form. A) 2665\frac { 26 } { 65 }6526​ B) 616792\frac { 616 } { 792 }792616​ C) 129215\frac { 129 } { 215 }215129​

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

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 = 22 . 103 . 711 . 135 Choices: 814212835\text { Choices: } 8 \quad 14 \quad 21 \quad 28 \quad 35 Choices: 814212835

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

Which numbers are prime? If a number is not prime, list at least three factors following the number. 3 5,231,211 61\cdot73 121 43

Is there a whole number M that would make this true? If so, tell what M is. If not, tell why not. A) 35 . 52 . 173 = 34 . 174 . M B) 24 . 72 . 118 . 22 = 25 . 7 . 116 . M

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.

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.

Name the number of factors of each of these numbers and list them, in factored form. A) 52 × 173 B) 35 ​

When the number 540 is written as a product of its prime factors in the form a2b3ca ^ { 2 } b ^ { 3 } ca2b3c , what is the numerical value of a + b + c? Choose one of the following:

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

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?

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

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.

Reasoning About Quantities

Numeration Systems

Understanding Whole Number Operations

Some Conventional Ways of Computing

Using Numbers in Sensible Ways

Meanings for Fractions

Computing With Fractions

Multiplicative Comparisons and Multiplicative Reasoning

Ratios, Rates, Proportions, and Percents

Integers and Other Number Systems

What Is Algebra

A Quantitative Approach to Algebra and Graphing

Understanding Change: Relationships Among Time, Distance, and Rate

Further Topics in Algebra and Change

Polygons

Polyhedra

Symmetry

Tessellations

Similarity

Curves, Constructions, and Curved Surfaces

Transformation Geometry

Measurement Basics

Area, Surface Area, and Volume

Counting Units Fast: Measurement Formulas

Special Topics in Measurement

Quantifying Uncertainty

Determining More Complicated Probabilities

Introduction to Statistics and Sampling

Representing and Interpreting Data With One Variable

Dealing With Multiple Data Sets or With Multiple Variables

Variability in Samples

Special Topics in Probability

Filters

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

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.

Write these numbers in simplest form. A) $\frac { 26 } { 65 }$ B) $\frac { 616 } { 792 }$ C) $\frac { 129 } { 215 }$

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 .} 10³^. 7^{11 .} 13⁵ $\text { Choices: } 8 \quad 14 \quad 21 \quad 28 \quad 35$

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

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.

Name the number of factors of each of these numbers and list them, in factored form. A) 5² × 17³ B) 3⁵

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:

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.