12-240/Homework Assignment 2 - Revision history

Drorbn at 20:40, 25 September 2012

2012-09-25T20:40:24Z

← Older revision		Revision as of 16:40, 25 September 2012
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	:You don't need to submit this exercise at all, but you will learn a lot by doing it!		:You don't need to submit this exercise at all, but you will learn a lot by doing it!

	* Add your name to the [[12-240/Class Photo\|Class Photo]]!		* Add your name to the [[12-240/Class Photo\|Class Photo]] page!

Drorbn at 20:16, 25 September 2012

2012-09-25T20:16:57Z

← Older revision		Revision as of 16:16, 25 September 2012
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	*# With <math>a</math> and <math>b</math> as in the previous two parts, show that <math>ba^{p-1}=b</math> in <math>{\mathbb F}_p</math>, and therefore <math>a^{p-1}=1</math> in <math>{\mathbb F}_p</math>.		*# With <math>a</math> and <math>b</math> as in the previous two parts, show that <math>ba^{p-1}=b</math> in <math>{\mathbb F}_p</math>, and therefore <math>a^{p-1}=1</math> in <math>{\mathbb F}_p</math>.
	*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?		*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?
	You don't need to submit this exercise at all, but you will learn a lot by doing it!		:You don't need to submit this exercise at all, but you will learn a lot by doing it!

	* Add your name to the [[12-240/Class Photo\|Class Photo]]!		* Add your name to the [[12-240/Class Photo\|Class Photo]]!

Drorbn at 19:59, 25 September 2012

2012-09-25T19:59:14Z

← Older revision		Revision as of 15:59, 25 September 2012
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	*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?		*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?
	You don't need to submit this exercise at all, but you will learn a lot by doing it!		You don't need to submit this exercise at all, but you will learn a lot by doing it!

			* Add your name to the [[12-240/Class Photo\|Class Photo]]!

Drorbn at 00:50, 25 September 2012

2012-09-25T00:50:06Z

← Older revision		Revision as of 20:50, 24 September 2012
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	{{12-240/Navigation}}		{{12-240/Navigation}}
	{{In Preparation}}

	This assignment is due at the tutorials on Thursday October 4. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday October 4. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 19:21, 24 September 2012

2012-09-24T19:21:05Z

← Older revision		Revision as of 15:21, 24 September 2012
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	This assignment is due at the tutorials on Thursday October 4. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorials on Thursday October 4. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

	Read sections 1.1 through 1.3 in our textbook, and solve the following problems:		Read sections 1.1 through 1.4 in our textbook, and solve the following problems:

	* Problems <u>3a</u> and 3bcd on page 6, problems <u>1</u>, 7, <u>18</u>, 19 and <u>21</u> on pages 14-16 and problems <u>8</u>, 9, 11 and <u>19</u> on pages 20-21. You need to submit only the underlined problems.		* Problems <u>3a</u> and 3bcd on page 6, problems <u>1</u>, <u>7</u>, <u>18</u>, 19 and <u>21</u> on pages 12-16 and problems <u>8</u>, 9, 11 and <u>19</u> on pages 20-21. You need to submit only the underlined problems.

	* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=46,655</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.		* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=46,655</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.

Drorbn at 22:56, 23 September 2012

2012-09-23T22:56:26Z

New page

{{12-240/Navigation}}
{{In Preparation}}

This assignment is due at the tutorials on Thursday October 4. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the teaching assistants will be having hard time deciphering your work they will give up and assume it is wrong.

Read sections 1.1 through 1.3 in our textbook, and solve the following problems:

* Problems 3a and 3bcd on page 6, problems 1, 7, 18, 19 and 21 on pages 14-16 and problems 8, 9, 11 and 19 on pages 20-21. You need to submit only the underlined problems.

* Note that the numbers <math>1^6-1=0</math>, <math>2^6-1=63</math>, <math>3^6-1=728</math>, <math>4^6-1=4,095</math>, <math>5^6-1=15,624</math> and <math>6^6-1=46,655</math> are all divisible by <math>7</math>. The following four part exercise explains that this is not a coincidence. But first, let <math>p</math> be some odd prime number and let <math>{\mathbb F}_p</math> be the field with p elements as defined in class.
*# Prove that the product <math>b:=1\cdot 2\cdot\ldots\cdot(p-2)\cdot(p-1)</math> is a non-zero element of <math>{\mathbb F}_p</math>.
*# Let <math>a</math> be a non-zero element of <math>{\mathbb F}_p</math>. Prove that the sets <math>\{1,2,\ldots,(p-1)\}</math> and <math>\{1a,2a,\ldots,(p-1)a\}</math> are the same (though their elements may be listed here in a different order).
*# With <math>a</math> and <math>b</math> as in the previous two parts, show that <math>ba^{p-1}=b</math> in <math>{\mathbb F}_p</math>, and therefore <math>a^{p-1}=1</math> in <math>{\mathbb F}_p</math>.
*# How does this explain the fact that <math>4^6-1</math> is divisible by <math>7</math>?
You don't need to submit this exercise at all, but you will learn a lot by doing it!