1 Marks Question

Question 11 Mark

If $a$ and $b$ are relatively prime numbers, then what is their $LCM$?

Answer

If $a$ and $b$ are relatively prime numbers, then their $LCM$ is $ab$.
In $LCM$ we choose all prime factor and if two numbers are relatively prime, so their factor is $a$ and $b$.
So, $LCM (a, b) = ab$

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Question 21 Mark

Write the condition to be satisfied by $q$ so that a rational number $\frac{\text{p}}{\text{q}}$ has a terminating decimal expansion.

Answer

In the rational number $\frac{\text{p}}{\text{q}},$ the factorization of denominator $q$ must be in form of $2^m× 5^n$ where $m$ and $n$ are non-negative integers.

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Question 31 Mark

What is a lemma?

Answer

A lemma is a proven statement which is used to prove other statement.

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Question 41 Mark

Given that $HCF (306. 657) = 9$, find $LCM (306, 657)$.

Answer

By the property that $LCM \times HCF =$ Product of the two numbers
$LCM \times 9 = 306 \times 657$
$LCM$ $=\frac{306\times 657}{9}= 22338$
Therefore $LCM (306, 657) = 22338.$

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Question 51 Mark

Determine the prime factorisation of the following positive integer:
$20570$

Answer

$20570 = 2 × 5 × 12^4× 17$

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Question 61 Mark

If $p$ and $q$ are two prime numbers, then what is their $LCM?$

Answer

If $p$ and $q$ are two primes, their $LCM$ will be their product.

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Question 71 Mark

Express the following integers as a product of its prime factors:
$468$

Answer

To Express: Each of the following numbers as a product of their prime factors.
$468 = 2^2× 3^2× 13$

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Question 81 Mark

Express the following integers as a product of its prime factors:
$945$

Answer

To Express: Each of the following numbers as a product of their prime factors.
$945 = 3^3× 5 × 7$

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Question 91 Mark

Express the following integers as a product of its prime factors:
$420$

Answer

To Express: Each of the following numbers as a product of their prime factors.
$420 = 2^2× 3 × 5 × 7$

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Question 101 Mark

If $a$ and $b$ are relatively prime numbers, then what is their $HCF?$

Answer

$a$ and $b$ are relatively prime numbers than their $HCF$ is $1$ because in $HCF$ we choose common prime factor and if two numbers are relatively prime then their common factor is $1$.
$HCF (a, b) = 1$

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Question 111 Mark

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

Answer

We know that $2$ is the smallest prime number and $4$ is the smallest composite number $HCF$ of $2$ and $4 = 2$

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Question 121 Mark

Determine the prime factorisation of the following positive integer:
$45470971$

Answer

$45470971 = 7^2× 13^2× 17^2× 19$

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Question 131 Mark

What is the total number of factors of a prime number?

Answer

Total number of factors of a prime number is $2$ i.e., $1$ and itself.

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Question 141 Mark

For what value of $n$, $2^n \times 5^n$ ends in $5$.

Answer

$2^n \times 5^n$
If $\mathrm{n}=0$, then $2^0 \times 5^0=1 \times 1=1$
If $\mathrm{n}=1$, then $2^1 \times 5^1=2 \times 5=10$
If $n=2$, then $2^2 \times 5^2=4 \times 25=100$
Thus, no value of $n, 2^n \times 5^n$ and in $5$ .

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Question 151 Mark

Write the sum of the exponents of prime factors in the prime factorisation of $98$.

Answer

$98 = 2 × 7 × 7 = 2^1× 7^2$
Sum of exponents $= 1 + 2 = 3$

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Question 161 Mark

What is an algorithm?

Answer

Algorithm: An algorithm is a series of well defined slips which gives a procedure for solving a type of problem.

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Question 171 Mark