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Prime Time - Co-prime numbers for safekeeping treasures

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Two numbers are said to be co-prime if they have only 11 as their common factor. This means their Highest Common Factor (HCF) is 11.

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Co-prime numbers do not need to be prime numbers themselves. For example, 88 and 99 are both composite numbers, but they are co-prime because their only common factor is 11.

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Any two consecutive natural numbers are always co-prime. For example, (14,15)(14, 15), (20,21)(20, 21), and (99,100)(99, 100) are all pairs of co-prime numbers.

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Two different prime numbers are always co-prime to each other, such as 77 and 1111.

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If a number is divisible by two co-prime numbers, then it is also divisible by their product. For example, if a treasure code is divisible by 33 and 55 (which are co-prime), it is also divisible by 3×5=153 \times 5 = 15.

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The Least Common Multiple (LCM) of two co-prime numbers is always equal to their product: LCM(a,b)=a×bLCM(a, b) = a \times b.

📐Formulae

HCF(a,b)=1HCF(a, b) = 1

LCM(a,b)=a×b(if a and b are co-prime)LCM(a, b) = a \times b \quad \text{(if a and b are co-prime)}

Common Factors={1}\text{Common Factors} = \{1\}

💡Examples

Problem 1:

Verify if the numbers 1818 and 3535 are co-prime so they can be used as unique identifiers for safekeeping treasures.

Solution:

Factors of 1818: 1,2,3,6,9,181, 2, 3, 6, 9, 18. Factors of 3535: 1,5,7,351, 5, 7, 35. The only common factor between 1818 and 3535 is 11.

Explanation:

Since the HCF(18,35)=1HCF(18, 35) = 1, the numbers 1818 and 3535 are co-prime.

Problem 2:

A security lock requires a code that is divisible by both 55 and 66. Using the property of co-prime numbers, find the smallest such non-zero code.

Solution:

First, we check if 55 and 66 are co-prime. Factors of 5={1,5}5 = \{1, 5\}. Factors of 6={1,2,3,6}6 = \{1, 2, 3, 6\}. Their HCF=1HCF = 1. The smallest number divisible by both is their LCMLCM. For co-prime numbers, LCM=5×6=30LCM = 5 \times 6 = 30.

Explanation:

Because 55 and 66 are co-prime, any number divisible by both must be a multiple of their product, 3030.

Problem 3:

Is the pair (15,21)(15, 21) co-prime?

Solution:

Factors of 15=1,3,5,1515 = 1, 3, 5, 15. Factors of 21=1,3,7,2121 = 1, 3, 7, 21. Common factors are 11 and 33.

Explanation:

Since there is a common factor other than 11 (which is 33), the HCFHCF is 33. Therefore, 1515 and 2121 are NOT co-prime.