Exercise 1.3
Q.1: Express each of the following integers as a product of its prime.
1. 420
2. 468
3. 945
4. 7325
Sol:
To express: each of the following numbers as a product of their prime factors
1. 420
420 = 2 x 2 x 3 x 5 x 7
2. 468
468 = 2 x 2 x 3 x 3 x 13
3. 945
945 = 3 x 3 x 3 x 5 x 7
4. 7325
7325 = 5 x 5 x 293
Q.2: Determine the prime factorization of each of the following positive integer :
1. 20570
2. 58500
3. 45470971
Sol:
TO EXPRESS: each of the following numbers as a product of their prime factors.
1. 20570
20570 = 2 x 5 x 11 x 11 x 17
2. 58500
58500 = 2 x 2 x 3 x 3 x 5 x 5 x 5 x 13
3. 45470971
45470971 = 7 x 7 x 13 x 13 x 17 x 17 x 19
Q.3: Explain why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers.
Sol:
Explanation:
Why 7 x 11 x 13 + 13 and 7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 are composite numbers.
We can see that both the numbers have common factor 7 and 1.
7 x 11 x 13 + 13 = (77 + 1) x 13 = 78 x 13
7 x 6 x 5 x 4 x 3 x 2 x 1 + 5 = (7 x 6 x 4 x 3 x 2 + 1) x 5 = 1008 x 5
And we know that composite numbers are those numbers which have at least one more factor other than 1.
Hence after simplification we see that both numbers are even and therefore the given two numbers are composite numbers
Q.4: Check whether 6n can end with the digit 0 for any natural number n.
Sol:
TO CHECK: Whether 6n can end with the digit 0 for any natural number n.
We know that 6n = (2 x 3)n
6n = 2n x 3n
Therefore, prime factorization of 6n does not contain 5 and 2 as a factor together. Hence 6n can never end with the digit 0 for any natural number n.