01204211/homework6 number theory 1 - ประวัติรุ่นแก้ไข

Jittat เมื่อ 16:09, 11 ตุลาคม 2558

2015-10-11T16:09:29Z

Jittat เมื่อ 16:08, 11 ตุลาคม 2558

2015-10-11T16:08:02Z

Jittat เมื่อ 01:30, 8 ตุลาคม 2558

2015-10-08T01:30:19Z

Jittat เมื่อ 01:29, 8 ตุลาคม 2558

2015-10-08T01:29:50Z

Jittat เมื่อ 02:17, 5 ตุลาคม 2558

2015-10-05T02:17:07Z

Jittat เมื่อ 02:15, 5 ตุลาคม 2558

2015-10-05T02:15:14Z

Jittat เมื่อ 02:43, 4 ตุลาคม 2558

2015-10-04T02:43:06Z

Jittat เมื่อ 02:23, 4 ตุลาคม 2558

2015-10-04T02:23:16Z

Jittat เมื่อ 02:22, 4 ตุลาคม 2558

2015-10-04T02:22:30Z

Jittat เมื่อ 02:20, 4 ตุลาคม 2558

2015-10-04T02:20:36Z

@@ แถว 34: / แถว 34: @@
 '''H.8''' (LPV-6.10.14) Find pairs of integers for which the Euclidean Algorithm runs for (a) 2 steps, and (b) 6 steps.
-''Notes:'' Recall that the Euclidean algorithm <tt>GCD(a,b)</tt> is defined as:
+''Notes:'' ''Recall that the Euclidean algorithm <tt>GCD(a,b)</tt> is defined as:''
   FUNCTION GCD(a,b)
@@ แถว 40: / แถว 40: @@
 .  return GCD(b, a mod b)
-In this problem let's count the number of steps the algorithm runs by the number of times function GCD is called.  For example, GCD(8,2) runs 1 one step, but GCD(18,10) runs for 3 steps.
+''In this problem let's count the number of steps the algorithm runs by the number of times function GCD is called.  For example, GCD(8,2) runs 1 one step, but GCD(18,10) runs for 3 steps.  In this problem, you only have to give '''examples''' of pairs for cases (a) and (b).''
 '''H.9''' (LPV-6.10.16) Prove that for every positive integer <math>m</math>, there is a Fibonacci number, beside <math>F_0=0</math>, divisible by <math>m</math>.   For example, when <math>m=4</math>, we have <math>F_6=8</math> which is divisible by <math>m</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 16:08, 11 ตุลาคม 2558
แถว 33:		แถว 33:

	'''H.8''' (LPV-6.10.14) Find pairs of integers for which the Euclidean Algorithm runs for (a) 2 steps, and (b) 6 steps.		'''H.8''' (LPV-6.10.14) Find pairs of integers for which the Euclidean Algorithm runs for (a) 2 steps, and (b) 6 steps.
		+
		+	''Notes:'' Recall that the Euclidean algorithm <tt>GCD(a,b)</tt> is defined as:
		+
		+	FUNCTION GCD(a,b)
		+	1. if b\|a then return b endif
		+	2. return GCD(b, a mod b)
		+
		+	In this problem let's count the number of steps the algorithm runs by the number of times function GCD is called. For example, GCD(8,2) runs 1 one step, but GCD(18,10) runs for 3 steps.


	'''H.9''' (LPV-6.10.16) Prove that for every positive integer <math>m</math>, there is a Fibonacci number, beside <math>F_0=0</math>, divisible by <math>m</math>. For example, when <math>m=4</math>, we have <math>F_6=8</math> which is divisible by <math>m</math>.		'''H.9''' (LPV-6.10.16) Prove that for every positive integer <math>m</math>, there is a Fibonacci number, beside <math>F_0=0</math>, divisible by <math>m</math>. For example, when <math>m=4</math>, we have <math>F_6=8</math> which is divisible by <math>m</math>.

@@ แถว 1: / แถว 1: @@
 : ''This is part of [[01204211-58]]''
-'''Due:''' TBA
+'''Due:''' 21 Oct, 2015
 '''H.1''' (LPV-6.1.3) Prove that if <math>a|b</math> and <math>a|c</math>, then <math>a|b+c</math> and <math>a|b-c</math>.

@@ แถว 6: / แถว 6: @@
-'''H.2''' Prove that for integers <math>a</math> and <math>b</math> and positive integer <math>q</math> the following statements are true.
+'''H.2''' (LPV-6.10.3) Prove that if <math>c\neq 0</math> and <math>ac|bc</math>, then <math>a|b</math>.
 (a) <math>(a+b)\;\bmod\; q = ((a\;\bmod\; q) + (b\;\bmod\; q))\;\bmod\; q</math>
@@ แถว 15: / แถว 18: @@
-'''H.3''' (LPV-6.1.6) (a) Prove that for every integer <math>a</math>, <math>a-1|a^2-1</math>.
+'''H.4''' (LPV-6.1.6) (a) Prove that for every integer <math>a</math>, <math>a-1|a^2-1</math>.
 (b) More generally, prove that for every integer <math>a</math> and positive integer <math>n</math>, <math>a-1|a^n-1</math>.
-'''H.4''' (LPV-6.3.3) Suppose that <math>a</math> and <math>b</math> are integers and <math>a|b</math>.  Suppose that <math>p</math> is a prime and <math>p|b</math>, but <math>p\not| a</math>.  Prove that <math>p|(b/a)</math>.
+'''H.5''' (LPV-6.3.3) Suppose that <math>a</math> and <math>b</math> are integers and <math>a|b</math>.  Suppose that <math>p</math> is a prime and <math>p|b</math>, but <math>p\not| a</math>.  Prove that <math>p|(b/a)</math>.
-'''H.7'''
+'''H.8''' (LPV-6.10.14) Find pairs of integers for which the Euclidean Algorithm runs for (a) 2 steps, and (b) 6 steps.
-'''H.8'''
+'''H.9''' (LPV-6.10.16) Prove that for every positive integer <math>m</math>, there is a Fibonacci number, beside <math>F_0=0</math>, divisible by <math>m</math>.   For example, when <math>m=4</math>, we have <math>F_6=8</math> which is divisible by <math>m</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:17, 5 ตุลาคม 2558
แถว 8:		แถว 8:
	'''H.2''' Prove that for integers <math>a</math> and <math>b</math> and positive integer <math>q</math> the following statements are true.		'''H.2''' Prove that for integers <math>a</math> and <math>b</math> and positive integer <math>q</math> the following statements are true.

−	(a) <math>(a+b)\;\bmod q = ((a\;\bmod q) + (b\;\bmod q))\;\bmod q</math>	+	(a) <math>(a+b)\;\bmod\; q = ((a\;\bmod\; q) + (b\;\bmod\; q))\;\bmod\; q</math>
		+
		+	(b) <math>(ab)\;\bmod\; q = ((a\;\bmod\; q) \times (b\;\bmod\; q))\;\bmod\; q</math>
		+
		+	Notes: this fact may be useful: If <math>r=a\;\bmod\;q</math>, then there exists an integer <math>k</math> such that <math>a=kq+r</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:15, 5 ตุลาคม 2558
แถว 4:		แถว 4:

	'''H.1''' (LPV-6.1.3) Prove that if <math>a\|b</math> and <math>a\|c</math>, then <math>a\|b+c</math> and <math>a\|b-c</math>.		'''H.1''' (LPV-6.1.3) Prove that if <math>a\|b</math> and <math>a\|c</math>, then <math>a\|b+c</math> and <math>a\|b-c</math>.
		+
		+
		+	'''H.2''' Prove that for integers <math>a</math> and <math>b</math> and positive integer <math>q</math> the following statements are true.
		+
		+	(a) <math>(a+b)\;\bmod q = ((a\;\bmod q) + (b\;\bmod q))\;\bmod q</math>

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:23, 4 ตุลาคม 2558
แถว 14:		แถว 14:


−	'''H.4''' Let <math>p</math> be a prime. Consider integers <math>a</math> and <math>b</math>, such that <math>1\leq a,b\leq p-1</math>. Prove that if <math>a \bmod p = b\bmod p</math> then <math>a=b</math>.	+	'''H.4''' Let <math>p</math> be a prime. Consider integers <math>a</math> and <math>b</math>, such that <math>1\leq a,b\leq p-1</math>. Prove that if <math>a \ \bmod p = b \ \bmod p</math> then <math>a=b</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:22, 4 ตุลาคม 2558
แถว 14:		แถว 14:


−	'''H.4'''	+	'''H.4''' Let <math>p</math> be a prime. Consider integers <math>a</math> and <math>b</math>, such that <math>1\leq a,b\leq p-1</math>. Prove that if <math>a \bmod p = b\bmod p</math> then <math>a=b</math>.

←รุ่นแก้ไขก่อนหน้า		รุ่นแก้ไขเมื่อ 02:20, 4 ตุลาคม 2558
แถว 11:		แถว 11:


−	'''H.3'''	+	'''H.3''' (LPV-6.3.3) Suppose that <math>a</math> and <math>b</math> are integers and <math>a\|b</math>. Suppose that <math>p</math> is a prime and <math>p\|b</math>, but <math>p\not\| a</math>. Prove that <math>p\|(b/a)</math>.