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Real Numbers: Theorems Euclid's

1. If a prime number p divides a squared integer a2, then p must also divide the integer a. 2. The numbers √2 and √3 are irrational, meaning they cannot be expressed as a ratio of two integers. 3. A rational number with a decimal expansion that terminates can be expressed as p/q where p and q are integers with no common factors other than 1, and q is of the form 2n5m for non-negative integers n and m.

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74 views1 page

Real Numbers: Theorems Euclid's

1. If a prime number p divides a squared integer a2, then p must also divide the integer a. 2. The numbers √2 and √3 are irrational, meaning they cannot be expressed as a ratio of two integers. 3. A rational number with a decimal expansion that terminates can be expressed as p/q where p and q are integers with no common factors other than 1, and q is of the form 2n5m for non-negative integers n and m.

Uploaded by

Tarun
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1.Let p be a prime number.

Division lemma
If p a divides a2, then p
divides a ,where a is a
Theorems Euclid’s Given positive
integers a and b,
positive integer. there exist unique
integers q and r
satisfying
2. √2, √3 are irrational Real Numbers a =bq + r ; 0 ≤ r < b

3. Let x be a rational number Division Algorithm


whose decimal expansion Prime
terminates. Then x can be Steps to obtain the
expressed in the form, p/q Factorization Fundamental HCF of two
where p & q are coprime, the Method Theorem Of positive integers,
prime factorisation of q is of the Arithmetic say c and d ,with
form 2n 5 m where n, m are non- c >d
negative integers.
Step 1: Apply
4. Let x =p/q be a rational
Euclid’s Division
number such that the prime For any two Every Lemma, to
factorisation of q is of the form Composite Composite
positive integers c & d. c =dq + r
2n 5 m where n, m are non- Number can Number
a and b be expressed
negative integers. Then, x has a x = P1×P2× Step 2: If r = zero
decimal expansion which HCF(a, b) X as a product
terminates. of primes, P3...×Pn , d is the HCF of
LCM(a, b)= a x b where c and d If r ≠ 0,
and this
For example factorisation P1P2 ... Pn apply Euclid's
is unique, Division to
5. Let x =p/q be a rational F(x)=3x2y apart from are
d and r
number, such that the prime g(x)=6xy2 the order in prime
factorisation of q is not of the which the numbers
form 2n 5 m of where n, m are HCF= 3xy prime factors Step 3: Continue
non-negative integers. Then, x occur the process till
has a decimal expansion which LCM=6x2y2
is non-terminating repeating the remainder is
zero

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