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Welcome to our Math lesson on Other Divisibility Rules. How Relatively Prime Numbers Determine the Divisibility Rules., this is the lesson of our suite of math lessons covering the topic of Divisibility Rules, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
By definition, relatively prime numbers are those numbers that don't have any other common factor besides 1.
For example, 2 and 3, 2 and 5, 3 and 5, etc., are relatively prime numbers, as they don't have any common factor besides 1.
These numbers may not necessarily be prime on themselves. For example, 9 and 16 are relatively prime despite neither of them being prime (9 is divisible by 3, while 16 is divisible by 2, 4 and 8 besides 1 and themselves). However, 9 and 16 do not have any common divisor except 1, hence they are relatively prime.
Relatively prime number are useful in determining other divisibility rules besides those discussed above. You may have noticed that the divisibility rules by numbers that are products of relatively prime numbers are similar - all of them require that the original number be divisible by both the prime factors. For example, divisibility by 6 implies the divisibility by both 2 and 3; divisibility by 12 implies the divisibility by both 3 and 4, divisibility by 15 implies the divisibility by both 3 and 5, and divisibility by 18 implies the divisibility by both 2 and 9. All these numbers are relatively prime.
Likewise, we deduce that the divisibility by 21 implies the divisibility by both 3 and 7; divisibility by 22 implies the divisibility by both 2 and 11; divisibility by 24 implies the divisibility by both 3 and 8; divisibility by 26 implies the divisibility by both 2 and 11; divisibility by 33 implies the divisibility by both 3 and 11 and so on.
Hence, in general, if a number is divisible by each of two relatively prime numbers, it is also divisible by their product.
Check whether 36,822 is divisible by 34 and 57 or not.
Divisibility by 34 requires that the number be divisible by both 2 and 17. It is clear that 36,822 is divisible by 2, as it is even. As for the divisibility by 17, we have 3682 - 5 × 2 = 3672; then 367 - 5 × 2 = 357; then 35 - 5 × 7 = 0, which means the original number is divisible by 17 and therefore by 34.
Divisibility by 57 requires that the original number be divisible by both 3 and 19. We have 3 + 6 + 8 + 2 + 2 = 21, which is divisible by 3, as 21 ÷ 3 = 7. On the other hand, 36,822 is also divisible by 19 because 2 × 2 + 3682 = 3686; then 2 × 6 + 368 = 380; then 2 × 0 + 38 = 38, which is a number divisible by 19 (38 ÷ 19 = 2).
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