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Back to: MBOSE Mathematics Class 10
Real Numbers
Real Numbers Euclid’ Division Lemma
Find HCF using Euclid’s division lemma
Euclid’s Division Lemma says thats given two positive integers a and b, there exist unique integers q and r such that
a=bq+r, 0≤r<b
The integer q is the quotient, and the integer r is the remainder. The quotient and remainder are unique. In simple words, Euclid’s Division Lemma says that whenever you divide an integer by another (non zero) integer, you will get a unique quotient and a unique remainder integer.
17=3(5)+2
24=7(3)+3
61=11(5)+6
7=12(0)+7
In each line, the underlined numbers are the dividend and divisor respectively, the number in brackets is the quotient, while the last number is the remainder. Note: The remainder is always less than the divisor.
Real Numbers #3: Solving problems using Euclid’s division lemma
Recap Of HCF and LCM
Solved Questions
Fundamental Theorem of Arithmetic
Solving problems using Fundamental Theorem of Arithmetic
Fundamental Theorem of Arithmetic:
Every composite number can be expressed (factorised ) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
This theorem also says that the prime factorisation of a natural number is unique, except for the order of its factors. For example 20 can be expressed as 2X2X5
Using this theorem the LCM and HCF of the given pair of positive integers can be calculated.
LCM = Product of the greatest power of each prime factor, involved in the numbers.
HCF = Product of the smallest power of each common prime factor in the numbers.
Prove that √2 is irrational number
Decimal representation of rational numbers in
terms of terminating/non – terminating recurring
decimals.
