# If two positive integers a and b are written as a = x3y2 and b = xy2 where x, y are prime numbers, then HCF (a, b) is (a) xy (b) xy2 (c) x3 y3 (d) x2 y2

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Updated on: September 25th, 2023

HCF (a, b) is xy2. The HCF is a crucial concept in number theory that helps us find the largest value that divides both numbers without leaving a remainder. By exploring the relationship between the prime factors of “a” and “b,” we can find the HCF of these two integers.

## Steps to Calculate HCF of a & b

Given that, a = x3y2 = x × x × x × y × y

b = xy2 = x × y × y

The product of each common prime facter’s smallest power yields the number HCF.

HCF of a and b = HCF (x3y2, xy2) = x × y × y = xy2

HCF (a, b) is xy2

## Integers

In mathematics, an integer is a grouping of positive and negative numbers. Like whole numbers, integers do not include the fractional component. Therefore, numbers that can be positive, negative, or zero but are not fractions are said to be integers. All mathematical operations, such as addition, subtraction, multiplication, and division, can be performed on integers. Z represents an integer.

## Types of Integers

Integers come in three types:

1. Zero (0) – Zero is not an integer, either positive or negative. There is no (+ or -) sign on zero, making it a neutral number.
2. Natural Numbers (Positive Integers) – Natural numbers, often known as counting numbers, are positive integers. The symbol Z+ is also used to represent these integers.
3. The negative of natural numbers are known as negative integers (additive inverse of natural numbers). The symbol for them is Z-. On a number line, the negative integers are located to the left of zero.

Summary:

## If two positive integers a and b are written as a = x3y2 and b = xy2 where x, y are prime numbers, then HCF (a, b) is

If two positive integers a and b are written as a = x3y2 and b = xy2 where x, y are prime numbers, then HCF (a, b) is xy2. HCF is the largest positive integer that divides two or more given numbers without leaving any remainder. It is determined by identifying the common prime factors of the numbers and finding the product of the lowest exponent for each common prime factor.

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