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# Find the greatest number which divides $399,434 \& 537$ leaving the remainder $8,9 \& 10$ respectively.  Verified
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Hint: Real number can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol $''R''$. All the natural numbers, decimals, and fractions come under this category. See the figure, given below, which shows the classification of real numbers.
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,

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]
‘$17$’ is the common and greatest number that divides $399,434\& 537$ completely.