a
Middle
School
Grade 6
Grade 7
Grade 8
Grade 9
Ressource
Out Middle
High
School
Grade 10
Grade 11
Grade 11 - STMG
Grade 12 - STMG
Grade 12
Grade 12 - Comp.
Grade 12 - Exp.
Ressource
Out High
4
3
2
1
Outside the high school program
/ Arithmetic
17 exercises (including 15 corrected)
a
Euclidean division
(2 exercices)
Pgcd with the Euclidean algorithm
(7 exercices)
Integers, PGCD, PPCM
(2 exercices)
Unlimited development
(1 exercice)
Pythagorean triplet
(1 exercice)
Bezout's theorem and Euclid's algorithm
(2 exercices)
image PNG : /home/_math/_exercice/d0/14/metapost/01.png
20
euro
20
20
s
0021291386
s
0021291386
Dividende
Diviseur
Reste
6452
3241
...
...
...
...
...
...
...
...
...
...
×
6452
...
3241
...
×
...
...
...
...
×
...
...
...
...
×
...
...
...
...
1.
Euclide
an
division
E.1718
W
e
giv
e
the
Euclidean
division
of
195
695
b
y
3
:
195695
=
65231
×
3
+
2
1
By
studying
the
pro
duct
(65231
×
3+2)
×
2
,
determine
the
Euclidean
division
of
195695
×
2
b
y
3.
2
Do
the
same
for
the
Euclidean
division
of
195
695
×
3
b
y
3.
3
Complete
the
follo
wing
table
:
n
1
2
3
4
5
6
7
Euclidean
division
of
195695
×
n
E.16
It
is
admitted
that
w
e
obtain
the
même
remainder
b
y
dividing
an
in
teger
b
y
9
than
b
y
dividing
the
sum
of
its
digits
b
y
9.
F
or
example
:
8753
=
972
×
9
+
5
the
remainder
is
therefore
5.
8
+
7
+
5
+
3
=
23
=
2
×
9
+
5
the
remainder
is
also
5.
Euro
banknotes
feature
an
11-digit
co
de
preceded
b
y
a
letter.
The
letter
is
replaced
b
y
its
place
in
the
usual
alphab
et
of
26
digits.
This
pro
duces
an
in
teger
with
12
or
13
digits,
and
w
e
find
the
remainder
of
dividing
this
n
um
b
er
b
y
9
.
This
re-
mainder
is
the
même
for
all
gen
uine
banknotes
and
is
w
orth
8
.
Example
:
Co
de
:
s
00
212
913
862
Rank
in
the
alphab
et
of
the
letter
s
:
19
Num
b
er
obtained
:
1
900
212
913
862
Remainder
for
this
banknote
:
8
1
The
co
de
u
01
308
937
097
app
ears
on
a
banknote.
a
Giv
e
the
13
-digit
in
teger
corresp
onding
to
this
co
de.
b
Calculate
the
remainder
of
the
division
b
y
9
of
the
sum
of
the
13
digits
of
this
in
teger.
c
What
can
w
e
sa
y
ab
out
this
banknote?
2
On
a
gen
uine
banknote
app
ears
the
co
de
s
02
166
448
10
x
,
x
for
the
last
illegible
digit.
Sho
w
that
x
+42
is
congruen
t
to
8
mo
dulo
9.
Deduce
x
.
3
On
another
authen
tic
banknote,
the
part
of
the
co
de
formed
b
y
the
11
digits
is
16
122
340
242
,
but
the
letter
preceding
them
has
b
een
erased.
W
e
call
n
the
p
osition
of
the
erased
letter
in
the
alphab
et.
a
Determine
the
p
ossible
v
alues
of
n
.
b
What
are
the
p
ossibilities
for
the
erased
letter??
2.
Pgc
d
with
the
Euclide
an
algorithm
E.13
T
o
mak
e
the
fraction
6452
3241
irreducible,
w
e
will
calculate
the
PGCD
of
the
in
tegers
6452
and
3241?
h
ttps:/
/c
hingmath.fr
sacados/1718
chapExoCorrec/16
sacados/16
20
euro
20
20
s
0021291386
s
0021291386
chapExoCorrec/13
sacados/13
Dividende
Diviseur
Reste
6452
3241
...
...
...
...
...
...
...
...
...
...
×
6452
...
3241
...
×
...
...
...
...
×
...
...
...
...
×
...
...
...
...
Dividende
Diviseur
Reste
240
36
...
...
...
...
...
...
...
×
240
...
36
...
×
...
...
...
...
×
...
...
...
...
Dividende
Diviseur
Reste
410
246
...
...
...
...
...
...
...
×
410
...
246
...
×
...
...
...
...
×
...
...
...
...
E.14
In
a
newly
built
house,
w
e
w
an
t
to
tile
the
flo
ors
in
certain
ro
oms.
1
The
dining
ro
om
flo
or
is
a
rectangle
of
length
4.54
m
and
width
3.75
m
.
W
e
w
an
t
to
tile
this
ro
om
with
square
tiles
33
cm
square.
W
e
start
the
installation
with
a
ˇ
c
orner
ı
of
the
ro
om
as
suggested
in
the
adjacen
t
figure.
Calculate
the
n
um
b
er
of
uncut
tiles
that
will
ha
v
e
b
een
laid.
2
The
kitc
hen
flo
or
is
a
rectangle
of
length
4.55
m
and
width
3.85
m
.
W
e
w
an
t
to
tile
this
ro
om
with
a
whole
n
um
b
er
of
square
tiles,
without
an
y
cutting.
a
Giv
e
the
list
of
divisors
of
455
and
then
the
list
of
divisors
of
385.
b
Giv
e
the
list
of
divisors
common
to
455
and
385.
c
What
is
then
the
largest
p
ossible
side
of
the
square
tiles
to
b
e
used
to
tile
this
kitc
hen?
3
W
e
ha
v
e
rectangular
tiles
of
length
24
cm
and
width
15
cm
.
a
Giv
e
the
list
of
m
ultiples
of
24
less
than
400,
then
the
list
of
m
ultiples
of
15
less
than
400
b
Giv
e
the
list
of
m
ultiples
common
to
24
and
15,
less
than
400.
c
What
w
ould
b
e
the
side
length
of
the
smallest
square
ro
om
that
could
b
e
tiled
with
an
in
teger
n
um
b
er
of
suc
h
tiles,
without
an
y
cutting?
E.3328
1
Using
Euclid’s
algorithm,
complete
the
table
b
elo
w
to
determine
the
PGCD
of
the
n
um
b
ers
240
and
36
:
2
Mak
e
the
fraction
240
36
irreducible
b
y
p
erforming
a
sin-
gle
simplification.
By
what
in
teger
ha
v
e
y
ou
simplified?
Justify
y
our
approac
h.
E.4995
1
Using
Euclid’s
algorithm,
determine
the
P
GC
D
of
the
in
tegers
410
and
246
:
2
a
Simplify
the
fraction
246
410
.
b
P
erform
the
follo
wing
calculations
:
246
410
−
8
5
;
1
246
−
1
410
E.3329
1
Calculate
the
Greatest
Common
Divisor
(PGCD)
of
496
and
806.
2
W
rite
496
806
as
an
irreducible
fraction.
3
Calculate
496
806
−
3
26
(we’l
l
give
the
r
esult
as
an
irr
e
ducible
fr
action)
E.3749
Using
Euclid’s
algorithm,
determine
the
P
GC
D
of
the
in
tegers
a
and
b
:
a
a
=
354
;
b
=
20
b
a
=
1
456
;
b
=
256
c
a
=
17
;
b
=
3941
d
a
=
256
419
;
b
=
3
866
E.6077
Consider
the
equation
(
E
)
:
44
·
x
+
35
·
y
=
2
x
∈
N
;
y
∈
N
1
a
Using
Euclid’s
algorithm,
sho
w
that
the
in
tegers
44
and
35
are
prime
to
eac
h
other.
b
Determine
a
pair
(
x
0
;
y
0
)
v
erifying
the
relation
:
44
·
x
0
+
35
·
y
0
=
1
2
Deduce
the
set
of
solutions
to
the
equation
(
E
)
.
3.
Inte
gers,
PGCD,
PPCM
E.259
1
Giv
e
the
pro
duct
decomp
osition
of
prime
factors
of
the
follo
wing
natural
in
tegers
:
a
25
×
72
b
54
×
12
c
32
×
84
d
100
×
98
2
F
rom
question
1
,
determine
:
a
The
PGCD
of
the
pair
(
25
×
72
;
54
×
12
)
;
b
The
PPCM
of
the
pair
(
32
×
84
;
100
×
98
)
of
natural
in-
tegers.
3
Use
a
divisor
tree
to
determine
the
set
of
divisors
of
the
in
teger
315.
E.277
1
Determine
the
decomp
ositions
in
to
prime
factors
of
the
follo
wing
in
tegers
:
A
36
×
54
b
125
×
134
c
280
×
24
2
Deduce
the
PGCD
and
PPCM
of
the
pair
(
6720
;
16750
)
of
natural
n
um
b
ers.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/14
sacados/14
chapExoCorrec/3328
sacados/3328
Dividende
Diviseur
Reste
240
36
...
...
...
...
...
...
...
×
240
...
36
...
×
...
...
...
...
×
...
...
...
...
chapExoCorrec/4995
sacados/4995
Dividende
Diviseur
Reste
410
246
...
...
...
...
...
...
...
×
410
...
246
...
×
...
...
...
...
×
...
...
...
...
chapExoCorrec/3329
sacados/3329
chapExoCorrec/3749
sacados/3749
chapExoCorrec/6077
sacados/6077
chapExoCorrec/259
sacados/259
chapExoCorrec/277
sacados/277
4.
Unlimite
d
development
E.1822
Consider
the
geometric
sequence
(
u
n
)
with
first
term
u
0
=
2
and
reason
3.
1
a
Determine
the
terms
u
1
,
u
2
,
u
3
and
u
4
.
b
Giv
e
the
base
7
writing
of
u
2
.
c
Sho
w
that
writing
in
base
7
is
105
7
.
d
T
o
obtain
the
base
7
writing
of
u
4
,
a
studen
t
has
p
er-
formed
the
m
ultiplication
b
elo
w.
Sa
y
whether
or
not
he
is
righ
t
and
explain
wh
y
.
1
0
5
×
3
3
1
5
2
a
Sho
w
that
:
u
5
=
486
.
b
Consider
the
function
f
,
tak
en
from
an
algorithm,
and
taking
as
argumen
t
a
a
natural
n
um
b
er
:
Function
f(a)
L
←
empty
list
x
←
a
As
long
as
x
>
0
r
←
remainder
of
the
division
euclidean
division
of
x
by
7
Add
r
at
the
top
of
the
list
L
x
←
q
End
While
Return
L
Complete
the
tableb
elo
w
to
record
the
differen
t
v
al-
ues
tak
en
b
y
the
v
ariables
of
function
f
when
calling
function
function
f
using
parameter
a
=
486
:
r
q
L
x
Initialisation
vide
486
End
of
stage
1
End
of
stage
2
.
.
.
.
.
.
Explain
the
link
b
et
w
een
the
items
in
the
list
L
and
the
writing
of
u
5
in
base
7.
3
W
e
divide
the
term
u
10
of
the
sequence
(
u
n
)
b
y
some
in-
teger.
W
e
obtain
the
quotien
t
Q
whose
decimal
writing
is
Q
=
14.727
272
727
272
72
:::
writing
in
whic
h
the
digits
7
and
2
rep
eat
to
infinit
y
.
Let
(
v
n
)
b
e
the
geometric
sequence
of
first
term
0.72
and
reason
0.01
.
a
Calculate
v
0
+
v
1
+
v
2
b
W
e
p
ose
S
n
=
v
0
+
v
1
+
v
2
+
···
+
c
n
where
n
is
a
non-zero
natural
n
um
b
er.
c
Calculate
S
n
.
Deduce
lim
n
↦→
+
∞
S
n
.
d
Deduce
a
writing
of
0.727
272
:
:
:
where
the
digits
7
and
2
rep
eat
to
infinit
y
in
the
form
of
the
quotien
t
of
t
w
o
in
tegers.
What
is
the
n
um
b
er
b
y
whic
h
w
e
divided
u
10
?
5.
Bezout’s
the
or
em
and
Euclid’s
algorithm
E.3751
F
or
eac
h
equation,
determine
a
pair
of
in
teger
solutions
(
u
;
v
)
:
a
354
·
u
+
49
·
v
=
1
b
34
·
u
+
57
·
v
=
1
E.3753
1
Determine
a
couple
(
x
;
y
)
of
in
tegers
solutions
of
the
equation
:
56
·
x
+
45
·
y
=
1
2
Deduce
a
couple
(
x
;
y
)
’of
relativ
e
in
tegers
v
erifying
the
equalit
y
:
56
·
x
+
45
·
y
=
3
6.
Unclassifie
d
financial
ye
ars
E.29
1
Without
the
calculator,
p
erform
the
Euclidean
division
of
372
b
y
15
.
2
With
the
calculator,
determine
the
Euclidean
division
of
37
852
b
y
23
.
3
a
2456
=
17
×
143
+
25
Deduce
the
Euclidean
division
of
2
456
b
y
17
.
b
32
247
143
=
225
+
72
143
Deduce
the
Euclidean
division
of
32
247
b
y
143
.
c
516
43
=
12
Deduce
the
Euclidean
division
of
516
b
y
43
.
d
674
24
=
28
+
1
12
Deduce
the
Euclidean
division
of
674
b
y
24
.
e
5460
63
=
86
+
2
3
Deduce
the
Euclidean
division
of
5
460
b
y
63
.
h
ttps:/
/c
hingmath.fr
sacados/1822
chapExoCorrec/3751
sacados/3751
chapExoCorrec/3753
sacados/3753
chapExoCorrec/29
sacados/29
E.336
W
e
w
an
t
to
determine
the
sign
of
the
follo
wing
expressions
:
2
y
;
x
+
y
;
x
−
y
;
x
(2
y
−
1)
;
−
x
+
y
In
the
follo
wing
cases
:
x
>
0
and
y
>
0
:
P
ositiv
e
Négatif
On
ne
p
eut
conclude
2
y
x
+
y
x
−
y
x
(2
y
−
1)
−
x
+
y
x
>
0
and
y
<
0
:
P
ositiv
e
Négatif
On
ne
p
eut
conclude
2
y
x
+
y
x
−
y
x
(2
y
−
1)
−
x
+
y
E.5302
The
aim
of
this
exercise
is
to
pro
v
e
b
y
absurdit
y
that
there
are
infinitely
man
y
prime
in
tegers
of
the
form
4
n
−
1
,
where
n
is
an
elemen
t
of
N
∗
(set
of
non-zer
o
nat-
ur
al
numb
ers)
.
1
Let
E
b
e
the
set
of
prime
in
tegers
of
the
form
4
n
−
1
where
n
is
an
elemen
t
of
N
∗
.
Sho
w
that
E
has
at
least
t
w
o
elemen
ts.
2
Assume
E
finite
.
Let
P
b
e
the
pro
duct
of
all
elemen
ts
of
E
and
X
=
4
P
−
1
.
a
Find
a
minoran
t
of
X
.
b
Sho
w
that
X
is
not
divisible
b
y
2
,
and
deduce
that
an
y
prime
factor
of
X
is
either
of
the
form
4
n
+1
,
or
of
the
form
4
n
−
1
where
n
is
an
elemen
t
of
N
∗
.
c
Sho
w
that
X
has
at
least
one
prime
factor
of
the
form
4
n
−
1
where
n
is
an
elemen
t
of
N
∗
.
3
Considering
a
prime
factor
p
of
X
of
the
form
4
n
−
1
,
the
definition
of
P
and
the
relation
X
=
4
P
−
1
,
complete
the
demonstration
b
y
the
absurd.
h
ttps:/
/c
hingmath.fr
chapExoCorrec/336
sacados/336
chapExoCorrec/5302
sacados/5302
ChingMath
by ChingLab
©
Middle School
Grade 6
Grade 7
Grade 8
Grade 9
High School
Grade 10
Grade 11
Grade 12
Grade 12 opt. Complémentaire
Grade 12 opt. Experte
Various
Contact me
F.A.Q
Other website
ChingWeb