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Rational = \[\dfrac{5}{3},0.63,0.0\overline {12} \]

Integers = \[\{ .....2, - 1,0,1,2....\} \]

Whole = \[\{ 0,1,2,3.....\} \]

Natural = \[\{ 1,2,3....\} \]

Irrational = \[\sqrt {3,} \sqrt {1,} 0.100100....\]

The fundamental theorem of Arithmetic-

This theorem states that every integer greater than 1 is either a prime number or can be expressed in the form of primes. To recall, Prime factors are numbers that are divisible by 1 and itself only.

Using this theorem the LCM and HCF of the given pair of positive integer can be calculated

LCM= Product of the greatest power of each prime factor, involving in the numbers.

HCF= Product of the smallest power of each common prime factor is in numbers.

Therefore,

Given,

The number divides \[399,434\& 537\] leaving the remainder \[8,9\& 10\].

So that number will divide \[(399 - 8),(434 - 9)\& (537 - 10)\] completely.

Now the greatest number that will divide \[399,434\& 537\] completely, is the HCF of these numbers.

\[391 = 17 \times 23\] [prime factor of 391]

\[425 = 5 \times 5 \times 17\][prime factor of 425]

\[527 = 17 \times 31\][prime factor of 527]

Hence,

â€˜\[17\]â€™ is the common and greatest number that divides \[399,434\& 537\] completely.

The product of the given number is equal to the product of their HCF and LCM. This result is true for all positive integers and is often used to find the HCF of the two given numbers if their LCM is given and vice versa.

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