问题1.将以下每个整数表示为其质数。
我)420
ii)468
iii)945
iv)7325
解决方案：
Let us express each of the numbers as a product of prime factors.
i) 420
Performing prime factorisation of the number, we get,
420 = 2 × 2 × 3 × 5 × 7
ii) 468
Performing prime factorisation of the number, we get,
468 = 2 × 2 × 3 × 3 × 13
iii) 945
Performing prime factorisation of the number, we get,
945 = 3 × 3 × 3 × 5 × 7
iv) 7325
Performing prime factorisation of the number, we get,
7325 = 5 × 5 × 293
问题2.确定以下每个正整数的质数分解：
我)20570
ii)58500
iii)45470971
解决方案：
Let us express each of the numbers as a product of prime factors.
i) 20570
Performing prime factorisation of the number, we get,
20570 = 2 × 5 × 11 × 11 × 17
ii) 58500
Performing prime factorisation of the number, we get,
58500 = 2 × 2 × 3 × 3 × 5 × 5 × 5 × 13
iii) 45470971
Performing prime factorisation of the number, we get,
45470971 = 7 × 7 × 13 × 13 × 17 × 17 × 19
问题3.解释为什么7×11×13 + 13和7×6×5×4×3×2×1 + 5是复合数字。
解决方案：
Both of these numbers have a common factor of 7. Also, every number is divisible by 1.
7 × 11 × 13 + 13 = (77 + 1) × 13 = 78 × 13
7 × 6 × 5 × 4 × 3 × 2 × 1 + 5 = (7 × 6 × 4 × 3 × 2 + 1) × 5 = 1008 × 5
Composite numbers are those numbers which have at least one more factor other than 1.
Now,
Both of these numbers are even. Therefore, the given two numbers are composite numbers
问题4.检查6n是否对于任何自然数n都可以以数字0结尾。
解决方案：
Since, 6n = (2 × 3)n
6n = 2n × 3n
Any number can end with 0 if it divisible by 10 or 5 and 2 together. The, prime factorisation of 6n does not contain 5 and 2 as a pair of factors.
Therefore, 6n can never end with the digit 0 for any natural number n.