Explain why 13233343563715 is a composite number?

1

(a)

The given number ends in 5. Hence it is a multiple of 5. Therefore it is a composite number

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then calculate the least prime factor of (a + b).

1

(a)

a and b are two positive integers such that the least prime factor of a is 3 and the least prime factor of b is 5. Then least prime factor of (a + b) is 2.

What is the HCF of the smallest composite number and the smallest prime number?

1

(a)

The smallest prime number is 2 and the smallest composite number is 2² Hence, required HCF (2²,2) = 2.

Calculate the HCF of 3³ x 5 and 3² x 5².

1

(a)

HCF of 3³ x 5 and 3² x 5²
= 3² x 5
= 9 x 5
= 45

If HCF (a, b) = 12 and a x b = 1,800, then find LCM (a, b).

1

(a)

We know that a x b = HCF (a, b) x LCM (a, b)
⇒ 1,800 = 12 x LCM (a, b)
⇒  LCM(a,b)=\(\frac { 1,800 }{ 12 } =150\)

What is the condition for the decimal expansion of a rational number to terminate? Explain with the help of an example.

1

(a)

The decimal expansion of a rational number terminates, if the denominator of rational no p/q, when p and q are co-primes and q can be expressed as 2^m5ⁿ  where In and n are non-negative integers.
e.g. \(\frac { 3 }{ 10 } =\frac { 3 }{ 2^{ 1 }\times 5^{ 1 } } =0.3\)

Find the smallest positive rational number by which 1/7 should be multiplied so that its decimal expansion terminates after 2 places of decimal.

1

(a)

Since ​​​​​​ ​\(\frac { 1 }{ 7 } +\frac { 7 }{ 100 } =\frac { 1 }{ 100 } =0.01\)
Thus smallest rational number is \(\\ \frac { 7 }{ 100 } \)

What type of decimal expansion does a rational number has? How can you distinguish it from decimal expansion of irrational numbers?

1

(a)

A rational number is either terminating or nonterminating repeating.
An irrational number is non-terminating and non-repeating

Calculate \(\frac { 3 }{ 8 } \) in the decimal form.

1

(a)

\(\frac { 3 }{ 8 } =\frac { 3 }{ { 2 }^{ 3 } } \)
 \(=\frac { { 3\times 5 }^{ 3 } }{ { 2 }^{ 3 }\times { 5 }^{ 3 } } \)
 \(=\frac { 375 }{ 10^{ 3 } } =\frac { 375 }{ 1000 } \)
 \(=0.375\)

The decimal representation of  \(\\ \frac { 6 }{ 1250 } \)  will terminate 1250 after how many places of decimal?

1

(a)

\(\frac { 6 }{ 1250 } =\frac { 6 }{ 2\times { 5 }^{ 4 } } \)
 \(=\frac { 6\times { 2 }^{ 3 } }{ 2\times { 2 }^{ 3 }\times { 5 }^{ 4 } } \)
 \(=\frac { 6\times { 2 }^{ 3 } }{ { 2 }^{ 4 }\times { 5 }^{ 4 } } \)
\(=\frac { 6\times { 2 }^{ 3 } }{ (10)^{ 4 } } =0.0288\)
\(\frac { 6 }{ 1250 } \) will terminate after 4 decimal places.