Sol.
Check whether 6n can end with the digit 0 for any natural number n.
Q.5
For e.g. 10, 20, 30,…
These nos. are divisible by 5
If the number 6n for any n ∈ N ends with the digit ‘0’,
then it is divisible by 5.
That means the prime factorisation of 6n must contain the prime number 5.
But,
6n = 6 × 6 × 6 × ….
That can also be written as,
6n = 2 × 3 × 2 × 3 × 2 × 3 × ….
It is not possible to get prime number 5
But this is not possible, because the primes in the prime factorisation of 6n are 2 and 3.
By Fundamental Theorem of Arithmetic
there are no other prime numbers except 2 and 3 in the factorisation of 6n.
So, there is no natural number for which 6n ends with the digit 0.
Exercise 1.2
Sol.
Q.6
7
×
11
×
+
=
13
(7
×
11
+
1)
=
13
(77
+
1)
=
13
×
=
13
×
13
×
2
×
3
Also,
7
×
6
×
×
4
×
3
×
2
×
1
+
=
5
(7
×
6
×
4
×
3
×
2
+
1)
=
5
(1008
+
1)
=
5
×
1009
Product of primes
Product of primes
Composite numbers are those numbers which can be expressed as product of primes
Explain why 7 × 11 × 13 + 13 and 7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 are composite numbers.
13
13
78
5
5
7 × 11 × 13 + 13 is a composite number
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 is a composite number.
Exercise 1.2