Prime Numbers, Factors and Multiples：练习题

Which of the following is the prime factorization of \( 126 \)?

Find the smallest positive integer \( n \) greater than 1 such that \( n \) is both a perfect square and a perfect cube.

A rectangular floor measures \(5.4\text{ m}\) by \(4.2\text{ m}\). It is to be covered with the largest possible square tiles of the same size so that the tiles fit exactly. How many tiles are needed in total?

Consider the number \( N = 2^{a} \times 3^{2} \times 5^{1} \). It is given that the number of positive factors of \( N \) is \( 24 \). Find the value of \( a \) and determine the smallest positive integer \( m \) such that \( N \times m \) is a perfect square.

The sum of three consecutive prime numbers is 49. What is the product of these three prime numbers?

Find the prime factorization of the number 180, expressing your answer in index notation.

Find the smallest positive integer that has exactly \( 10 \) factors.

Determine the smallest three-digit number that is a multiple of \( 9 \) and has exactly \( 6 \) factors.

A large rectangular container measures \(210\text{ cm}\) by \(252\text{ cm}\) by \(420\text{ cm}\).
(a) Express \(210\), \(252\), and \(420\) as products of their prime factors.
(b) Find the Highest Common Factor (HCF) of these three dimensions.
(c) If the container is to be filled completely with identical cubes of the largest possible size, find the side length of each cube and the total number of cubes required.

Let \( n \) be a positive integer.
(a) If \( n \) has exactly \( 3 \) factors, what can you conclude about the nature of \( n \)? Provide an example.
(b) Find the smallest positive integer that has exactly \( 8 \) factors, where one of the factors is \( 6 \).
(c) List all the factors of the number found in part (b) and identify which of them are prime numbers.