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Byju's Answer
Standard X
Mathematics
Fundamental Theorem of Arithmetic
n1111+n55+n33...
Question
n
11
11
+
n
5
5
+
n
3
3
+
62
165
n
is a positive integer for all n ∈ N.
Open in App
Solution
Let P(n) be the given statement.
Now,
P
(
n
)
:
n
11
11
+
n
5
5
+
n
3
3
+
62
165
n
is
a
positive
integer
for
all
n
∈
N
.
Step
1
:
P
(
1
)
=
1
11
+
1
5
+
1
3
+
62
165
=
15
+
33
+
55
+
62
165
=
165
165
=
1
It
is
certainly
a
positive
integer
.
Hence
,
P
(
1
)
is
true
.
Step
2
:
Le
t
P
(
m
)
be
true
.
Then
,
m
11
11
+
m
5
5
+
m
3
3
+
62
165
m
is
a
positive
integer
.
Now
,
let
m
11
11
+
m
5
5
+
m
3
3
+
62
165
m
=
λ
,
where
λ
∈
N
is
a
positive
integer
.
We
have
to
show
that
P
(
m
+
1
)
is
true
whenever
P
(
m
)
is
true
.
To
prove
:
(
m
+
1
)
11
11
+
(
m
+
1
)
5
5
+
(
m
+
1
)
3
3
+
62
165
(
m
+
1
)
is
a
positive
integer
.
Now
,
(
m
+
1
)
11
11
+
(
m
+
1
)
5
5
+
(
m
+
1
)
3
3
+
62
165
(
m
+
1
)
=
1
11
m
11
+
11
m
10
+
55
m
9
+
165
m
8
+
330
m
7
+
462
m
6
+
462
m
5
+
330
m
4
+
165
m
3
+
55
m
2
+
11
m
+
1
+
1
5
m
5
+
5
m
4
+
10
m
3
+
10
m
2
+
5
m
+
1
+
1
3
m
3
+
3
m
2
+
3
m
+
1
+
62
165
m
+
62
165
=
m
11
11
+
m
5
5
+
m
3
3
+
62
165
m
+
m
10
+
5
m
9
+
15
m
8
+
30
m
7
+
42
m
6
+
42
m
5
+
31
m
4
+
17
m
3
+
8
m
2
+
3
m
+
1
11
+
1
5
+
1
3
+
6
105
=
λ
+
m
10
+
5
m
9
+
15
m
8
+
30
m
7
+
42
m
6
+
42
m
5
+
31
m
4
+
17
m
3
+
8
m
2
+
3
m
+
1
It
is
a
positive
integer
.
Thus
,
P
(
m
+
1
)
is
true
.
By
the
principle
of
mathematical
induction
,
P
(
n
)
is
true
for
all
n
∈
N
.
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