Find the longest tape that can measure the room dimensions of Length 825 cm, Breadth 675 cm, and Height 450 cm

Find the longest tape that can measure the room dimensions of Length 825 cm, Breadth 675 cm, and Height 450 cm

To find the longest tape that can measure the room dimensions of length 825 cm, breadth 675 cm, and height 450 cm, use the concept of the Highest Common Factor (HCF).

The HCF, or Greatest Common Divisor (GCD), is the largest number that divides each of the given numbers without leaving a remainder. It helps determine the longest tape needed that can evenly measure all three dimensions.

To find the HCF of 825, 675, and 450, start by breaking down each number into its prime factors:

Prime factorization of 825 = 3 x 5² x 11

Prime factorization of 675 = 3³ x 5²

Prime factorization of 450 = 2 x 3² x 5²

The HCF is determined by multiplying the smallest power of all common prime factors present in all three numbers.

In this case, Common factors are 3 and 5²

3 is common to all three numbers

5² = 25 is also common to all three numbers

Other primes like 2 and 11 are not common to all

HCF = 3 x 5² = 3 x 25 = 75

Therefore, the length of the longest tape required to measure the dimensions of the room is 75 cm

Concept of Number Theory

Number theory is a branch of mathematics that deals with the properties and relationships of numbers, particularly integers. It is often called "higher arithmetic" and is fundamental to many areas of both pure and applied mathematics.

Prime Numbers

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Prime numbers are the building blocks of integers, as every integer greater than 1 can be factored into a product of prime numbers. This is known as the Fundamental Theorem of Arithmetic. The process of breaking down a number into its prime components. This concept is essential for various mathematical procedures and problem-solving techniques.

Greatest Common Divisor (GCD)

The greatest common divisor of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder. It is used to simplify fractions and solve problems involving ratios and proportions.

Least Common Multiple (LCM)

The least common multiple is the smallest positive integer that is divisible by each of the given numbers. It is useful in solving problems where synchronization or repeated cycles are involved.

Applications

• Number theory algorithms are essential in data compression, error detection and correction, and many other computational processes.
• Ensures secure communication in banking, internet transactions, and confidential communications.
• Patterns and sequences derived from number theory are used to create musical scales and rhythms.
• Number theory provides essential tools and concepts that are widely used across various fields of mathematics and science. Its principles help in solving complex problems, understanding the structure of numbers, and developing secure systems in technology and cryptography.