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4
3
2
1
Grade 12 - Exp.
/ Annals of congruence
10 exercises (100% corrected)
a
1.
Unclassifie
d
financial
ye
ars
E.3142
Reminder
:
F
or
t
w
o
relativ
e
in
tegers
a
and
b
,
w
e
sa
y
that
a
is
congru-
en
t
to
b
mo
dulo
7
,
and
w
e
write
a
≡
b
(
mo
d.
7)
when
there
exists
a
relativ
e
in
teger
k
suc
h
that
a
=
b
+7
k
.
1
This
question
constitutes
an
organized
return
of
kno
wl-
edge
:
a
Let
a
,
b
,
c
and
d
b
e
relativ
e
in
tegers.
Sho
w
that
:
If
a
≡
b
(
mo
d.
7)
and
c
≡
d
(
mo
d.
7)
then
a
·
c
≡
b
·
d
(
mo
d.
7)
.
b
Deduce
that
:
for
a
and
b
non-zero
in
tegers.
If
a
≡
b
(
mo
d.
7)
then
for
an
y
natural
n
um
b
er
n
:
a
n
≡
b
n
(
mo
d.
7)
.
2
F
or
a
=
2
then
for
a
=
3
,
determine
a
non-zero
natural
n
um
b
er
n
suc
h
that
:
a
n
≡
1
(
mo
d.
7)
.
3
Let
a
b
e
a
natural
n
um
b
er
not
divisible
b
y
7
.
a
Sho
w
that
:
a
6
≡
1
(
mo
d.
7)
.
b
W
e
call
order
of
a
(
mo
d.
7)
,
and
w
e
denote
b
y
k
,
the
smallest
non-zero
natural
n
um
b
er
suc
h
that
a
k
≡
1
(
mo
d.
7)
.
Sho
w
that
the
remainder
r
of
the
Euclidean
division
of
6
b
y
k
v
erifies
a
r
≡
1
(
mo
d.
7)
.
Deduce
that
k
divides
6.
What
are
the
p
ossible
v
alues
of
k
?
c
Giv
e
the
order
mo
dulo
7
of
all
in
tegers
a
b
et
w
een
2
and
6.
4
T
o
an
y
natural
in
teger
n
,
w
e
asso
ciate
the
n
um
b
er
:
A
n
=
2
n
+
3
n
+
4
n
+
5
n
+
6
n
.
Sho
w
that
:
A
2006
≡
6
(
mo
d.
7)
E.3187
Giv
en
a
natural
n
um
b
er
n
2
,
w
e
prop
ose
to
in
v
estigate
the
existence
of
three
natural
n
um
b
ers
x
,
y
and
z
suc
h
that
:
x
2
+
y
2
+
z
2
≡
2
n
−
1
(
mo
d.
2
n
)
P
art
A
:
Study
of
t
w
o
sp
ecial
cases
1
In
this
question,
it
is
assumed
that
n
=
2
.
Sho
w
that
1
,
3
and
5
satisfy
the
previous
question.
2
In
this
question,
w
e
assume
n
=
3
.
a
Let
m
b
e
a
natural
n
um
b
er.
Cop
y
and
complete
the
table
b
elo
w
giving
the
remainder
r
of
the
Euclidean
di-
vision
of
m
b
y
8
and
the
remainder
R
of
the
Euclidean
division
of
m
2
b
y
8.
r
0
1
2
3
4
5
6
7
R
b
Can
w
e
find
three
natural
n
um
b
ers
x
,
y
and
z
suc
h
that
:
x
2
+
y
2
+
z
2
≡
7
(
mo
d.
8)
?
P
art
B
:
Study
of
the
general
case
where
n
3
Supp
ose
there
are
three
natural
n
um
b
ers
x
,
y
and
z
suc
h
that
:
x
2
+
y
2
+
z
2
≡
2
n
−
1
(
mo
d.
2
n
)
1
Justify
the
fact
that
the
three
natural
n
um
b
ers
x
,
y
and
z
are
all
o
dd
or
that
t
w
o
of
them
are
ev
en.
2
It
is
assumed
that
x
and
y
are
ev
en
and
that
z
is
o
dd.
W
e
then
p
ose
:
x
=
2
q
;
y
=
2
r
;
z
=
2
s
+
1
where
q
,
r
,
s
are
natural
n
um
b
ers.
a
Sho
w
that
:
x
2
+
y
2
+
z
2
≡
1
(
mo
d.
4)
.
b
Deduce
a
con
tradiction.
3
Assume
x
,
y
,
z
are
o
dd.
a
Pro
v
e
that,
for
an
y
non-zero
natural
n
um
b
er
k
,
k
2
+
k
is
divisible
b
y
2
.
b
Deduce
that
:
x
2
+
y
2
+
z
2
≡
3
(
mo
d.
8)
.
c
Conclude.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3142
sacados/3142
chapExoCorrec/3187
sacados/3187
E.3240
P
art
A
Let
N
b
e
a
natural,
o
dd,
non-prime
in
teger.
Assume
N
=
a
2
−
b
2
où
a
and
b
are
t
w
o
natural
n
um
b
ers.
1
Sho
w
that
a
and
b
do
not
ha
v
e
the
same
parit
y
.
2
Sho
w
that
N
can
b
e
written
as
the
pro
duct
of
t
w
o
natural
n
um
b
ers
p
and
q
.
3
What
is
the
parit
y
of
p
and
q
?
P
art
B
W
e
admit
that
250
507
is
not
prime.
W
e
prop
ose
to
lo
ok
for
pairs
of
natural
in
tegers
(
a
;
b
)
v
erifying
the
relation
:
(
E
)
:
a
2
−
250
507
=
b
2
1
Let
X
b
e
a
natural
n
um
b
er.
a
Giv
e
in
a
table,
the
p
ossible
remainders
of
X
mo
dulo
9
;
then
those
of
X
2
mo
dulo
9
.
b
Kno
wing
that
a
2
−
250
507
=
b
2
,
determine
the
p
ossible
remainders
mo
dulo
9
of
a
2
−
250
507
;
deduce
the
p
ossi-
ble
remainders
mo
dulo
9
of
a
2
.
c
Sho
w
that
the
p
ossible
remainders
mo
dulo
9
of
a
are
1
and
8
.
2
Justify
that
if
the
pair
(
a
;
b
)
v
erifies
the
relation
(
E
)
,
then
a
501
.
Sho
w
that
there
is
no
solution
of
the
t
yp
e
(
501
;
b
)
.
3
It
is
assumed
that
the
pair
(
a
;
b
)
v
erifies
the
relation
(
E
)
.
a
Demonstrate
that
a
is
congruen
t
to
503
or
to
505
mo
d-
ulo
9
.
b
Determine
the
smallest
natural
n
um
b
er
k
suc
h
that
the
pair
(
505+9
k
;
b
)
is
a
solution
of
(
E
)
,
then
giv
e
the
corresp
onding
solution
couple.
P
art
C
1
Deduce
from
the
previous
parts
a
writing
of
250
507
as
a
t
w
o-factor
pro
duct.
2
Are
the
t
w
o
factors
prime
to
eac
h
other?
3
Is
this
writing
unique?
E.3319
1
a
Determine
according
to
the
v
alues
of
the
non-zero
natural
n
um
b
er
n
the
remainder
in
the
Euclidean
divi-
sion
b
y
9
of
7
n
.
b
Then
demonstrate
that
:
2005
2005
≡
7
(
mo
d.
9)
.
2
a
Demonstrate
that
for
an
y
non-zero
natural
n
um
b
er
n
:
10
n
≡
1
(
mo
d.
9)
.
b
W
e
denote
b
y
N
a
natural
n
um
b
er
written
in
base
ten,
w
e
call
S
the
sum
of
its
digits.
Pro
v
e
the
follo
wing
relationship
:
N
≡
S
(
mo
d.
9)
.
c
Deduce
that
N
is
divisible
b
y
9
if,
and
only
if,
S
is
divisible
b
y
9
.
3
Assume
A
=
2005
2005
;
denote
b
y
:
B
the
sum
of
the
digits
of
A
;
C
the
sum
of
the
digits
of
B
;
D
the
sum
of
the
digits
of
C
.
a
Demonstrate
the
follo
wing
relationship
:
A
≡
D
(
mo
d.
9)
.
b
Kno
wing
that
2005
<
10000
,
demonstrate
that
A
is
written
in
decimal
n
umeration
with
at
most
8020
dig-
its.
Deduce
that
:
B
72180
.
c
Demonstrate
that
:
C
45
.
d
By
studying
the
list
of
in
tegers
less
than
45,
determine
a
ma
joran
t
of
D
smaller
than
15.
e
Pro
v
e
that
:
D
=
7
.
E.3554
Throughout
the
exercise,
n
de-
notes
a
non-zero
natural
n
um
b
er.
1
a
F
or
1
n
6
,
calculate
the
remainders
of
the
Eu-
clidean
division
of
3
n
b
y
7
.
b
Demonstrate
that,
for
an
y
n
,
3
n
+6
−
3
n
is
divisible
b
y
7
.
Deduce
that
3
n
and
3
n
+6
ha
v
e
the
same
remainder
in
division
b
y
7.
c
Using
the
previous
results,
calculate
the
remainder
of
the
Euclidean
division
of
3
1
000
b
y
7
.
d
In
general,
ho
w
can
w
e
calculate
the
remainder
of
the
Euclidean
division
of
3
n
b
y
7
,
for
an
y
n
?
e
Deduce
that,
for
an
y
natural
n
um
b
er
n
,
3
n
is
prime
with
7
.
2
Let
U
n
=
1+3+3
2
+
···
+3
n
−
1
=
n
−
1
i
=0
3
i
,
où
n
is
a
natural
n
um
b
er
greater
than
or
equal
to
2.
a
Sho
w
that
if
U
n
is
divisible
b
y
7
then
3
n
−
1
is
divisible
b
y
7.
b
Recipro
cally
,
sho
w
that
if
3
n
−
1
is
divisible
b
y
7
then
U
n
is
divisible
b
y
7
.
Deduce
the
v
alues
of
n
suc
h
that
U
n
is
divisible
b
y
7
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3240
sacados/3240
chapExoCorrec/3319
sacados/3319
Antilles-Guyane Juin 2005 5 points
chapExoCorrec/3554
sacados/3554
E.3572
W
e
prop
ose
to
determine
the
cou-
ples
(
n
;
m
)
of
non-zero
natural
n
um
b
ers
v
erifying
the
rela-
tion
:
7
n
−
3
×
2
m
=
1
(
F
)
1
W
e
assume
m
4
.
Sho
w
that
there
are
exactly
t
w
o
solution
pairs.
2
It
is
no
w
assumed
that
m
5
.
a
Sho
w
that
if
the
couple
(
n
;
m
)
v
erifies
the
relation
(
F
)
then
:
7
n
≡
1
(
mo
d.
32)
.
b
By
studying
the
remainders
of
division
b
y
32
of
p
o
w
ers
of
7,
sho
w
that
if
thecouple
(
n
;
m
)
v
erifies
the
relation
(
F
)
then
n
is
divisible
b
y
4
.
c
Deduce
that
if
the
couple
(
n
;
m
)
v
erifies
the
relation
(
F
)
then
:
7
n
≡
1
(
mo
d.
5)
d
F
or
m
5
,
are
there
pairs
(
n
;
m
)
of
natural
n
um
b
ers
v
erifying
the
relation
(
F
)
?
3
Conclude,
i.e.
determine
the
set
of
pairs
of
non-zero
nat-
ural
in
tegers
v
erifying
the
relation
(
F
)
.
E.3629
W
e
call
(
E
)
the
set
of
natural
n
um
b
ers
that
can
b
e
written
in
the
form
9+
a
2
où
a
is
a
non-
zero
natural
n
um
b
er
;
for
example
:
10
=
9
+
1
2
;
13
=
9
+
2
2
;
.
.
.
In
this
exercise,
w
e
prop
ose
to
study
the
existence
of
elemen
ts
of
(
E
)
that
are
p
o
w
ers
of
2
,
3
or
5
.
1
Study
the
equation
of
unkno
wn
a
:
a
2
+
9
=
2
n
où
a
∈
N
,
n
∈
N
,
n
4
.
a
Sho
w
that
if
a
exists,
a
is
o
dd.
b
Reasoning
mo
dulo
4
,
sho
w
that
the
prop
osed
equation
has
no
solution.
2
Study
the
equation
with
unkno
wn
a
:
a
2
+
9
=
3
n
où
a
∈
N
,
n
∈
N
,
n
3
.
a
Sho
w
that
if
n
3
,
3
n
is
congruen
t
to
1
or
3
mo
dulo
4
.
b
Sho
w
that
if
a
exists,
it
is
ev
en
and
deduce
that
neces-
sarily
n
is
ev
en.
c
W
e
p
ose
n
=
2
p
où
p
is
a
natural
n
um
b
er,
2
.
Deduce
from
a
factorization
of
3
n
−
a
2
,
that
the
prop
osed
equa-
tion
has
no
solution.
3
Study
the
equation
with
unkno
wn
a
:
a
2
+
9
=
5
n
où
a
∈
N
,
n
∈
N
,
n
2
.
a
Reasoning
mo
dulo
3
,
sho
w
that
the
equation
has
no
solution
if
n
is
o
dd.
b
W
e
p
ose
n
=
2
p
,
dra
wing
inspiration
from
2
c
sho
w
that
there
exists
a
single
natural
n
um
b
er
a
suc
h
that
a
2
+9
is
an
in
teger
p
o
w
er
of
5
.
E.5863
P
art
A
Consider
the
function
f
from
an
algorithm
où
its
argumen
ts
tak
e
non-zero
natural
in
tegers
as
v
alues
:
Function
f(a;b)
c
←
0
As
long
as
a
>
b
c
←
c+1
a
←
a
−
b
End
As
long
as
Return
(c;a)
1
Indicating
the
v
alues
of
the
v
ariables
tak
en
successiv
ely
when
calling
the
function
f
with
the
v
alues
a
=
13
and
b
=
4
.
2
Ho
w
to
in
terpret,
according
to
the
v
alues
a
and
b
sup-
plied
as
argumen
ts,
the
v
alue
of
the
pair
returned
b
y
the
function
f
on
a
call.
P
art
B
F
or
eac
h
letter
of
the
alphab
et,
use
the
table
b
elo
w
to
assign
an
in
teger
b
et
w
een
0
and
25
.
a
b
c
d
e
f
g
h
i
j
k
l
m
0
1
2
3
4
5
6
7
8
9
10
11
12
n
o
p
q
r
s
t
u
v
w
x
y
z
13
14
15
16
17
18
19
20
21
22
23
24
25
A
co
ding
pro
cess
is
defined
as
follo
ws
:
Step
1
:
T
o
the
letter
w
e
w
an
t
to
co
de,
w
e
asso
ciate
the
in
teger
m
corresp
onds
in
the
table.
Step
2
:
W
e
calculate
the
remainder
of
the
Euclidean
di-
vision
of
9
m
+
5
b
y
26
and
note
it
p
.
Step
3
:
T
o
the
in
teger
p
,
w
e
asso
ciate
the
corresp
onding
letter
in
the
table.
1
Co
de
letter
U
.
2
Mo
dify
the
function
f
of
the
algorithm
of
part
A
so
that
at
a
v
alue
of
m
en
tered
b
y
the
user,
it
returns
the
v
alue
of
p
,
calculated
using
the
previous
co
ding
pro
cedure.
P
art
C
1
Find
an
in
teger
x
suc
h
that
:
9
x
≡
1
(
mo
d.
26)
.
2
Then
demonstrate
the
equiv
alence
:
9
m
+5
≡
p
(
mo
d.
26)
⇐
⇒
m
≡
3
p
−
15
(
mo
d.
26)
3
Then
deco
de
the
letter
B
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/3572
sacados/3572
chapExoCorrec/3629
sacados/3629
Asie Juin 2004
chapExoCorrec/5863
sacados/5863
E.5960
1
Sho
w
that,
for
an
y
natural
n
um
b
er
n
,
2
3
n
−
1
is
a
m
ulti-
ple
of
7
(r
e
asoning
by
r
e
curr
enc
e
may
b
e
use
d)
.
Deduce
that
2
3
n
+1
−
2
is
a
m
ultiple
of
7
and
that
2
3
n
+2
−
4
is
a
m
ultiple
of
7
.
2
Determine
the
remainders
of
the
division
b
y
7
of
the
p
o
w-
ers
of
2
.
3
The
n
um
b
er
p
b
eing
a
natural
n
um
b
er,
consider
the
in-
teger
:
Q
p
=
2
p
+
2
2
p
+
2
3
p
a
If
p
=
3
n
,
what
is
the
remainder
of
the
divisioin
of
A
p
b
y
7
?
b
Sho
w
that
if
p
=
3
n
+1
then
A
p
is
divisible
b
y
7
.
c
Study
the
case
où
p
=
3
n
+2
4
Consider
the
in
tegers
a
and
b
written
in
the
binary
sys-
tem
:
a
=
1
001
001
000
;
b
=
1
000
100
010
000
Chec
k
that
these
t
w
o
n
um
b
ers
are
n
um
b
ers
of
the
form
A
p
.
Are
they
divisible
b
y
7
?
E.6795
P
arts
A
and
B
can
b
e
treated
indep
enden
tly
.
P
art
A
In
order
to
encrypt
a
message,
affine
encryption
is
used.
Eac
h
letter
of
the
alphab
et
is
asso
ciated
with
an
in
teger
as
sho
wn
in
the
table
b
elo
w
:
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z
0
1
2
3
4
5
6
7
8
9
1
0
1
1
1
2
1
3
1
4
1
5
1
6
1
7
1
8
1
9
2
0
2
1
2
2
2
3
2
4
2
5
Let
x
b
e
the
in
teger
asso
ciated
with
the
letter
to
b
e
co
ded.
W
e
determine
the
remainder
y
of
the
Euclidean
division
of
7
x
+5
b
y
26
,
then
w
e
deduce
the
letter
asso
ciated
with
y
(it
is
it
that
c
o
des
the
original
letter)
.
Example
:
M
corresp
onding
to
x
=
12
7
×
12
+
5
=
89
No
w
89
≡
11
(
mo
d.
26)
and
11
corresp
onding
to
the
letter
L
,
so
the
letter
M
is
enco
ded
b
y
the
letter
L
.
1
Co
de
letter
L
.
2
a
Let
k
b
e
a
relativ
e
in
teger.
Sho
w
that
if
k
≡
7
x
(
mo
d.
26)
then
15
·
k
≡
x
(
mo
d.
26)
.
b
Demonstrate
the
recipro
cal
of
the
previous
implica-
tion.
c
Deduce
that
y
≡
7
x
+5
(
mo
d.
26)
is
equiv
alen
t
to
x
≡
15
·
y
+3
(
mo
d.
26)
.
3
Using
the
previous
question
deco
de
the
letter
E
.
P
art
B
Consider
the
sequences
a
n
and
b
n
suc
h
that
a
0
and
b
0
are
in
tegers
b
et
w
een
0
and
25
inclusiv
e
and
for
an
y
natural
n
um
b
er
n
:
a
n
+1
=
7
·
a
n
+
5
b
n
+1
=
15
·
b
n
+
3
Sho
w
that
for
an
y
natural
n
um
b
er
n
:
a
n
=
a
0
+
5
6
×
7
n
−
5
6
F
or
the
rest
of
the
problem,
w
e
admit
that
for
an
y
natural
n
um
b
er
n
:
b
n
=
b
0
+
3
14
×
15
n
−
3
14
P
art
C
Decrypting
a
message
enco
ded
with
an
affine
cipher
p
oses
no
difficult
y
(we
c
an
test
312
p
airs
of
p
ossible
c
o
efficients)
.
T
o
in-
crease
this
decryption
difficult
y
,
w
e
prop
ose
to
use
a
k
ey
that
will
indicate
for
eac
h
letter
the
n
um
b
er
of
times
où
applies
the
affine
encryption
of
the
A
part
to
it.
F
or
example,
to
enco
de
the
w
ord
MA
TH
with
the
k
ey
2
−
2
−
5
−
6
,
w
e
apply
ˇ2ı
times
the
affine
cipher
to
the
letter
M
(this
gives
E
)
,
ˇ2ı
times
the
cipher
to
the
letter
A
,
ˇ5ı
times
the
cipher
to
the
letter
T
and
finally
ˇ6ı
times
the
cipher
to
the
letter
H
.
In
this
part,
w
e’ll
use
the
k
ey
2
−
2
−
5
−
6
.
Deco
de
the
letter
Q
in
the
w
ord
I
Y
Y
Q
.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/5960
sacados/5960
Polynesie Juin 1999
chapExoCorrec/6795
sacados/6795
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