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1....In picture
2....The fundamental theorem of Arithmetic(FTA) was proved by Carl Friedrich Gauss in the year 1801. It states that every composite number can be expressed as a product of prime numbers, this factorization is unique except for the order in which the prime factors occur.
3..... Rational number :
Rational numbers are numbers which can be expressed as a fraction and also as positive numbers, negative numbers and zero.
Example: 3/2 is a rational number. It means integer 3 is divided by another integer 2.
Irrational number :
The numbers which are not a rational number are called irrational numbers.
Example: √8=2.828…
4....According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r < b.
Let a be the positive odd integer which when divided by 6 gives q as quotient and r as remainder.
A/c to euclid devision lemma---:
a = bq + r
a = 6q + r………………….(1)
where, (0 ≤ r < 6)
So r can be either 0, 1, 2, 3, 4 and 5.
Case 1:
If r = 1, then equation (1) becomes
a = 6q + 1
The Above equation will be always as an odd integer.
Case 2 :
If r = 3, then equation (1) becomes
a = 6q + 3
The Above equation will be always as an odd integer.
Case 3:
If r = 5, then equation (1) becomes
a = 6q + 5
The above equation will be always as an odd integer.
∴ Any odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.
Hence prove
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