Let G be the set of all integers x that can be written asthe sum of the squares of two integers, x = a 2 + b 2 . Forexample, 13 = 3 2 + 2 2 is in G, while 7 fails to be in G. a. Listall integers x ≤ 10 thatare in G. b. Show that G is closed under multiplication: If x = a 2 + b 2 and y = c 2 + d 2 are in G. then so is their productxy. Hint: Consider the matrices [ a − b b a ] , [ c − d d c ] , their product, and their determinants. c. Given that 2642 = 31 2 + 41 2 and 3218 = 37 2 + 43 2 ,write 8 , 501 , 956 = 2642 ⋅ 3218 as the sum ofthe squares of two positive integers. You may usetechnology.
Let G be the set of all integers x that can be written asthe sum of the squares of two integers, x = a 2 + b 2 . Forexample, 13 = 3 2 + 2 2 is in G, while 7 fails to be in G. a. Listall integers x ≤ 10 thatare in G. b. Show that G is closed under multiplication: If x = a 2 + b 2 and y = c 2 + d 2 are in G. then so is their productxy. Hint: Consider the matrices [ a − b b a ] , [ c − d d c ] , their product, and their determinants. c. Given that 2642 = 31 2 + 41 2 and 3218 = 37 2 + 43 2 ,write 8 , 501 , 956 = 2642 ⋅ 3218 as the sum ofthe squares of two positive integers. You may usetechnology.
Solution Summary: The author calculates the integers in G such that xle 10 are 1,2,4,5,8,7,10.
Let G be the set of all integers x that can be written asthe sum of the squares of two integers,
x
=
a
2
+
b
2
. Forexample,
13
=
3
2
+
2
2
is in G, while 7 fails to be in G. a. Listall integers
x
≤
10
thatare in G. b. Show that G is closed under multiplication: If
x
=
a
2
+
b
2
and
y
=
c
2
+
d
2
are in G. then so is their productxy. Hint: Consider the matrices
[
a
−
b
b
a
]
,
[
c
−
d
d
c
]
, their product, and their determinants. c. Given that
2642
=
31
2
+
41
2
and
3218
=
37
2
+
43
2
,write
8
,
501
,
956
=
2642
⋅
3218
as the sum ofthe squares of two positive integers. You may usetechnology.
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RELATIONS-DOMAIN, RANGE AND CO-DOMAIN (RELATIONS AND FUNCTIONS CBSE/ ISC MATHS); Author: Neha Agrawal Mathematically Inclined;https://www.youtube.com/watch?v=u4IQh46VoU4;License: Standard YouTube License, CC-BY