a. Prove that 10 n ≡ ( − 1 ) n ( m o d 11 ) for every positive integer n . b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a 0 - a 1 + a 2 - ⋅ ⋅ ⋅ + ( − 1 ) n a n , when z is written in the form as described in the previous problem. a. Prove that 10 n ≡ 1 ( m o d 9 ) for every positive integer n . b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9 . (Hint: Any integer can be expressed in the form a n 10 n + a n − 1 10 n − 1 + ⋅ ⋅ ⋅ + a 1 10 + a 0 where each a i is one of the digits 0 , 1 , . . . , 9. )
a. Prove that 10 n ≡ ( − 1 ) n ( m o d 11 ) for every positive integer n . b. Prove that a positive integer z is divisible by 11 if and only if 11 divides a 0 - a 1 + a 2 - ⋅ ⋅ ⋅ + ( − 1 ) n a n , when z is written in the form as described in the previous problem. a. Prove that 10 n ≡ 1 ( m o d 9 ) for every positive integer n . b. Prove that a positive integer is divisible by 9 if and only if the sum of its digits is divisible by 9 . (Hint: Any integer can be expressed in the form a n 10 n + a n − 1 10 n − 1 + ⋅ ⋅ ⋅ + a 1 10 + a 0 where each a i is one of the digits 0 , 1 , . . . , 9. )
Solution Summary: The author explains the formula used to prove that n is a positive integer.
a. Prove that
10
n
≡
(
−
1
)
n
(
m
o
d
11
)
for every positive integer
n
.
b. Prove that a positive integer
z
is divisible by
11
if and only if
11
divides
a
0
-
a
1
+
a
2
-
⋅
⋅
⋅
+
(
−
1
)
n
a
n
, when
z
is written in the form as described in the previous problem.
a. Prove that
10
n
≡
1
(
m
o
d
9
)
for every positive integer
n
.
b. Prove that a positive integer is divisible by
9
if and only if the sum of its digits is divisible by
9
. (Hint: Any integer can be expressed in the form
a
n
10
n
+
a
n
−
1
10
n
−
1
+
⋅
⋅
⋅
+
a
1
10
+
a
0
where each
a
i
is one of the digits
0
,
1
,
.
.
.
,
9.
)
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