Question

show 1/root3 is an irrational no.
explain why 1259735 is a composite no.
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Answer 1

Step-by-step explanation:

1. Que:- Show that 1/√3 is an irrational number.

Sol:- Let 1/√3 is a rational number.

Here, 1/√3 = a/b where a and b are co - prime.

So, √3b = a

Squaring both sides,

(√3b)² = a²

So, 3b² = a² -------(1)

So, a² is dividble by 3.

That means a is also dividble by 3.

Now, a = 3c (where c is natural number)

Now, again squaring both sides.

a² = 3²c²

Put equation 1 in a²

3b² = 3²c²

Cut 3 with 3³, we get,

b² = 3²c²

That means b² is dividble by 3. So, b also.

but at starting we took that a and b are co - prime. So, it contradicts our supposition.

So, 1/√3 is an irrational number.

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2. Que:- Why 1259735 is composite number?

Sol:- It's factors are 1 , 3 , 5 , 1259735 and so on. Hence, it has more than 2 factors. So, it is composite number.

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EXTRA INFORMATION:-

• Rational numbers are the numbers which are in the form of p/q where p and q are natural numbers and q ≠ 0. For example:- √25 , √36 , 2 , 4 ⁰/1 and so on.
• Irrational numbers are the numbers which have non terminating and non repeating decimal expansion. For example:- √3 , √7 , 2.67467699432356 and so on.
• Co - prime numbers are the numbers which never cut on any table, they are consecutive. Like, say, 2 , 3 and 5, 6 and so on.
• Composite numbers are the numbers which have more than 2 factors. Like , 4 , 6 , 8 , 9 and so on.

Answer 2

Step-by-step-explanation:-

:- Let 1/√3 is a rational number.

Here, 1/√3 = a/b where a and b are co - prime.

So, √3b = a

Squaring both sides,

(√3b)² = a²So, 3b² = a² -------(1)So, a² is dividble by 3.

That means a is also dividble by 3.

Now, a = 3c (where c is natural number)

Now, again squaring both sides.

a² = 3²c²

Put equation 1 in a²

3b² = 3²c²Cut 3 with 3³,

we get,b² = 3²c²

That means b² is dividble by 3.

So, b also.

But at starting we took that a and b are co - prime. So, it contradicts our supposition.So, 1/√3 is an irrational number.

2. Sol:- It's factors are 1 , 3 , 5 , 1259735 and so on. Hence, it has more than 2 factors. So, it is composite number.

:- It's factors are 1 , 3 , 5 , 1259735 and so on. Hence, it has more than 2 factors. So, it is composite number._____________________