Question

# Find the greatest number which divides $399,434 \& 537$ leaving the remainder $8,9 \& 10$ respectively.

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,

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

Note: Many arithmetic functions are defined using the Canonical representation. In particular, the value of additive and multiplicative functions are determined by their value on the powers of prime numbers.
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.