Egyptian mathematics had a unique way of writing fractions as sums of unit fractions – that is, fractions of the form 1 n . For example, the fraction 2 9 could be written as 1 6 + 1 18 and also as 1 5 + 1 45 . They would not represent a number like 2 3 as 1 3 + 1 3 ; they would use two different unit fractions. Unit fractions were written by placing the symbol , which looked somewhat like an eye, over the numeral. For example, 1 3 could be written as . In exercises 79 − 82 , write the given fractions as the sum of unit fractions, using our common notation rather than the cumbersome Egyptian notation. (There may be several correct answers but we will give only one.) 2 3
Egyptian mathematics had a unique way of writing fractions as sums of unit fractions – that is, fractions of the form 1 n . For example, the fraction 2 9 could be written as 1 6 + 1 18 and also as 1 5 + 1 45 . They would not represent a number like 2 3 as 1 3 + 1 3 ; they would use two different unit fractions. Unit fractions were written by placing the symbol , which looked somewhat like an eye, over the numeral. For example, 1 3 could be written as . In exercises 79 − 82 , write the given fractions as the sum of unit fractions, using our common notation rather than the cumbersome Egyptian notation. (There may be several correct answers but we will give only one.) 2 3
Solution Summary: The author explains how to write the tion 23 as a sum of unit tions by finding the prime factors of the denominator.
Egyptian mathematics had a unique way of writing fractions as sums of unit fractions – that is, fractions of the form
1
n
. For example, the fraction
2
9
could be written as
1
6
+
1
18
and also as
1
5
+
1
45
. They would not represent a number like
2
3
as
1
3
+
1
3
; they would use two different unit fractions. Unit fractions were written by placing the symbol , which looked somewhat like an eye, over the numeral. For example,
1
3
could be written as. In exercises
79
−
82
, write the given fractions as the sum of unit fractions, using our common notation rather than the cumbersome Egyptian notation. (There may be several correct answers but we will give only one.)
-
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
p-1
2
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
23
32
how come?
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
The set T is the subset of these residues exceeding
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2
multiple of n, i.e.
n mod p, 2n mod p, ...,
2
p-1
-n mod p.
Let T be the subset of S consisting of those residues which exceed p/2.
Find the set T, and hence compute the Legendre symbol (7|23).
The first 11 multiples of 7 reduced mod 23 are
7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8.
23
The set T is the subset of these residues exceeding
2°
So T = {12, 14, 17, 19, 21}.
By Gauss' lemma (Apostol Theorem 9.6),
(7|23) = (−1)|T| = (−1)5 = −1.
how come?
Shading a Venn diagram with 3 sets: Unions, intersections, and...
The Venn diagram shows sets A, B, C, and the universal set U.
Shade (CUA)' n B on the Venn diagram.
U
Explanation
Check
A-
B
Q Search
田
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