Which is the smallest number evenly divisible by all numbers from 0 to 10?

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2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder.

What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Least Common Multiple(LCM):

In arithmetic and number theory, the least common multiple, lowest common multiple, or smallest common multiple of two integers a and b, usually denoted by lcm, is the smallest positive integer that is divisible by both a and b.

Example:

LCM(6,7,21)

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

Multiples of 7: 7, 14, 21, 28, 35, 42, 56, 63

Multiples of 21: 21, 42, 63

Find the smallest number that is on all of the lists. We have it in bold above.

So LCM(6, 7, 21) is 42

The best solution with Euclid GCD algorithm

Euclid’s GCD algorithm:

1- The smallest positive number that is evenly divided (divided without remainder) by a > set of numbers is called the Least Common Multiple (LCM).

2- All we have to do to solve this problem is find the LCM for the integers {1, 2, 3, 4, > …, 20} using Euclid’s GCD algorithm.

3- After some reflection you might correctly realize that every integer is divisible by > > 1, so 1 can be removed from the list and use 2 through 20 instead.

4- we can eliminate other factors as well.

5- We leave 20 in the calculation but then remove its factors {2, 4, 5, 10}. Any number > > evenly divisible by 20 is also evenly divisible by these factors.

6- 19 is prime and has no factors— it stays.

7- 18 has factors {2, 3, 6, 9} and we already removed 2 but we can remove 3, 6, and 9. 17 > is prime — it stays.

8- We continue this numeric carnage until our original list of {1…20} becomes the much > > smaller {11…20}.

9- In general, all smaller factors of a bigger number in the list can be safely removed > > without changing the LCM.

Finding the factors of a number is an important math skill for basic arithmetic, algebra and calculus. The factors of a number are any numbers that divide into it exactly, including 1 and the number itself. In other words, every number is the product of multiple factors.

TL;DR (Too Long; Didn't Read)

The quickest way to find the factors of a number is to divide it by the smallest prime number (bigger than 1) that goes into it evenly with no remainder. Continue this process with each number you get, until you reach 1.

Prime Numbers

A number that can only be divided by 1 and itself is called a prime number. Examples of prime numbers are 2, 3, 5, 7, 11 and 13. The number 1 is not considered a prime number because 1 goes into everything.

Divisibility Rules

Some divisibility rules can help you find the factors of a number. If a number is even, it's divisible by 2, i.e. 2 is a factor. If a number's digits total a number that's divisible by 3, the number itself is divisible by 3, i.e. 3 is a factor. If a number ends with a 0 or a 5, it's divisible by 5, i.e. 5 is a factor.

If a number is divisible twice by 2, it's divisible by 4, i.e. 4 is a factor. If a number is divisible by 2 and by 3, it's divisible by 6, i.e. 6 is a factor. If a number is divisible twice by 3 (or if the sum of the digits is divisible by 9), then it's divisible by 9, i.e. 9 is a factor.

Finding Factors Quickly

Establish the number you want to find the factors of, for example 24. Find two more numbers that multiply to make 24. In this case, 1 x 24 = 2 x 12 = 3 x 8 = 4 x 6 = 24. This means the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24.

Factor negative numbers in the same way as positive numbers, but make sure the factors multiply together to produce a negative number. For example, the factors of -30 are -1, 1, -2, 2, -3, 3, -5, 5, -6, 6, -10, 10, -15 and 15.

If you have a large number, it's more difficult to do the mental math to find its factors. To make it easier, create a table with two columns and write the number above it. Using the number 3784 as an example, start by dividing it by the smallest prime factor (bigger than 1) that goes into it evenly with no remainder. In this case, 2 x 1892 = 3784. Write the prime factor (2) in the left column and the other number (1892) in the right column.

Continue with this process, i.e. 2 x 946 = 1892, adding both numbers to the table. When you reach an odd number (e.g., 2 x 473 = 946), divide by small prime numbers besides 2 until you find one that divides evenly with no remainder. In this case, 11 x 43 = 473. Continue the process until you reach 1.

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References

• Lamar State College-Orange: How to Find the Factors of a Number

Tips

• Practice practice practice! Soon you'll be able to find factors faster than ever!

About the Author

Claire is a writer and editor with 18 years' experience. She writes about science and health for a range of digital publications, including Reader's Digest, HealthCentral, Vice and Zocdoc.

Which number is divisible by all the numbers from 1 to 10?

Hence 2520 is the least number that is divisible by all the numbers between 1 and 10 (both inclusive)

Is a number ending in 0 always divisible by 10?

Any number ending in 0 is divisible by 10. If all of the digits of a number are the same and the number of digits is even, the number is divisible by 11. For example, 11 divides 66 and 2222.

Which is the smallest divisible number?

If you observe carefully the ans must be the LCM of the numbers 1 to n. Initialize ans = 1.

Which numbers are divisible by 10?

Divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10. Example: 10, 20, 30, 1000, 5000, 60000, etc.