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Find the smallest number which when increased by 20 is exactly divisible by 90 and 144.

Explore the perfect solution: Find the smallest number that, when increased by 20, becomes evenly divisible by both 90 and 144.

by Maivizhi A

Updated Mar 06, 2024

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Find the smallest number which when increased by 20 is exactly divisible by 90 and 144.

Find the smallest number which when increased by 20 is exactly divisible by 90 and 144 

The smallest number is 700.

To find the smallest number which when increased by 20 is exactly divisible by 90 and 144, we need to find the least common multiple (LCM) of 90 and 144.

First, we find the LCM of 90 and 144:

Prime factorization of 90: 90 = 2 * 3^2 * 5

Prime factorization of 144: 144 = 2^4 * 3^2

To find the LCM, we take the highest power of each prime factor present in the factorizations:

LCM(90, 144) = 2^4 * 3^2 * 5 = 720

Now, let x be the smallest number which when increased by 20 is exactly divisible by 720.

So, the smallest number is given by: x + 20 = 720n, where n is an integer.

To minimize x, we let n = 1:

x + 20 = 720 x = 720 - 20 x = 700

Therefore, the smallest number is 700. 

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Real Life Applications of LCM and HCF 

The concepts of Least Common Multiple (LCM) and Highest Common Factor (HCF), also known as Greatest Common Divisor (GCD), have numerous real-life applications across various fields including mathematics, computer science, engineering, finance, and more. Here are some practical applications:

  1. Fraction Operations: LCM is extensively used in adding, subtracting, multiplying, and dividing fractions. It helps in finding a common denominator and simplifying fractions.
  2. Scheduling and Time Management: LCM is used in scheduling tasks or events that recur at regular intervals. For example, finding the LCM of different time periods helps in scheduling meetings, appointments, or events that repeat cyclically.
  3. Repetitive Tasks: In industries where tasks are repeated at regular intervals, such as manufacturing or assembly lines, LCM helps in determining the least amount of time needed for all tasks to synchronize before they repeat.
  4. Electrical Engineering: In electrical circuits, LCM is used to determine the least common multiple of different frequencies or periods, aiding in synchronization of signals or designing circuits with repeating patterns.
  5. Computer Science and Information Technology: LCM and HCF are fundamental concepts used in algorithm design and optimization. They are used in various algorithms including cryptography, hashing, scheduling, and optimizing memory allocation.
  6. Optimization Problems: In optimization, finding the LCM or HCF can be a crucial step in minimizing or maximizing certain parameters. For example, in production planning, LCM can be used to find the least common multiple of different production cycles to optimize resource utilization.
  7. Finance: In finance, LCM and HCF are used in calculations involving interest rates, loan repayments, and investments. For instance, HCF can be used to simplify fractions representing interest rates or to find the common divisor for simplifying loan terms.
  8. Biology and Genetics: LCM and HCF are used in genetics to understand patterns of inheritance, particularly in determining the probabilities of certain genetic traits occurring in offspring.
  9. Telecommunications: LCM is used in telecommunications for scheduling data transmissions and optimizing bandwidth allocation.
  10. Project Management: LCM and HCF are used in project management to determine the timing and duration of various project tasks, ensuring efficient resource allocation and timely completion.

In essence, LCM and HCF are fundamental mathematical concepts with a wide range of practical applications across different domains, playing a crucial role in problem-solving and decision-making processes.

 

Find the smallest number which when increased by 20 is exactly divisible by 90 and 144 - FAQs

1. What is the smallest number when increased by 20 is exactly divisible by 90 and 144?

The smallest number is 700.

2. How do you find the smallest number satisfying the given conditions?

By finding the least common multiple (LCM) of 90 and 144.

3. What is the LCM of 90 and 144?

The LCM of 90 and 144 is 720.

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