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QUESTION: 1

Without actually performing the long division, state whether the following rational numbers will have a terminating decimal expansion or a non-terminating repeating decimal expansion.

Solution:
Denominator is 8, so the rational has a non-terminating decimal expansion

QUESTION: 2

Find the HCF of the following numbers: 18, 60.

Solution:
18, 60

18 = 1, 2, 3, 6, 9, 18

60 = 1, 2, 3, 4, 6, 10, 12, 15, 30, 60

Common factors = 1, 2, 3, 6

∴ H.C.F.of 18 & 60 is 6

QUESTION: 3

Find the HCF and LCM of the following integers by applying the prime factorization method 40, 36 and 126.

Solution:
40 = 2 x 2 x 2 x 5 = 23 x 5

36 = 2 x 2 x 3 x 3 = 22 x 32

and 126 = 2 x 3 x 3 x 7 = 2 x 32 x 7

For H.C.F taking minimum power of common factor H.C.F = (2)1 = 2

For L.C.M, taking maximum power of prime factors LC.M = 23 x 32 x 5 x 7 = 8 x 9 x 5 x 7 = 2520

Hence, H.C.F = 2 L.C.M = 2520

QUESTION: 4

Find the HCF by prime factorization 45 and 75.

Solution:
HCF of two numbers are

45 = 3 × 3 × 5

75 = 3 × 5 × 5

The common factors make the HCF: HCF = 3 × 5 = 15

QUESTION: 5

Find the H.C.F. of the following numbers using the prime factorization method: 84, 120, 138.

Solution:
84 = 2^{2} × 3 × 7

120 = 2^{3} × 3 × 5

138 = 2 × 3 × 23

HCF = product of common terms with lowest power

HCF = 2 × 3 = 6

QUESTION: 6

Check whether 6 / 200 has terminating or non terminating repeating decimal expansion.

Solution:
6 / 200 = 3 / 100 = 0.03 which is terminating decimal.

QUESTION: 7

Find the H.C.F of the following numbers by Prime Factorization: 120, 240, 300

Solution:
240 = 2^{4} x 3 x 5

300 = 2^{2} x 3 x 5^{2}

120 = 2^{3} x 3 x 5

H.C.F = 2^{2} x 3 x 5

Highest Common Factor is : 60

QUESTION: 8

Find the H.C.F. of the following numbers using the prime factorization method: 225, 450.

Solution:
We have, 225 = 3^{2} × 5^{2}

450 = 2 × 3^{2} × 5^{2}

HCF(225, 450) = Product of common terms with lowest power HCF(225, 450) = 3^{2} × 5^{2} HCF(225, 450) = 9 × 25

HCF(225, 450) = 225

∴ The required number or HCF is 225.

QUESTION: 9

If HCF of 336 and 54 is 6, then LCM is:

**Explanations:** According to Euclid division algorithm

LCM => HCF x LCM = product if two no. 6 x LCM = 54 x 336

LCM = 54 x 336 / 6

LCM = 3024

Solution:

QUESTION: 10

Prime factors of 4050 is:

Solution:
Thus, these are the numbers that make up the prime factorization of 4050,

So the prime factorization of 4050 is 2 × 3 × 3 × 3 × 3 × 5 × 5.

QUESTION: 11

The LCM of three numbers 28, 44, 132 is:

Solution:
Prime factors of 28 = 2^{2} x 7

Prime factors of 44 = 2^{2} x 11

Prime factors of 132 = 2^{2} x 3 x 11

LCM = 2^{2} x 7^{1} x 11^{1} x 3^{1} = 924

QUESTION: 12

If p^{2} is an even integer, them p is an:

Solution:
"Even" and "odd" are properties of integers, not of arbitrary real numbers. The correct statement is "if p is an integer and p^{2} is even, then p is even."

QUESTION: 13

Every composite number can be expressed as a product of:

Solution:
This is true as any composite number can be expressed in terms of its prime factors. The prime factors for 2 numbers can be the same but their combination is unique.

**Example:** 12 = 2^{2} x 3 and 18 = 2 x 3^{2}

Both have prime factors as 2 and 3 but the combination that 12 has is different from that of 18.

QUESTION: 14

The decimal expansion of a rational number is always:

Solution:
The decimal expansion of a rational number always either terminates after finitely many digits or begins to repeat the same sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number.

QUESTION: 15

The largest number which divides 445, 572 and 699 leaving remainder 4, 5 and 6 respectively is :

Solution:
445 - 4 = 441

572 - 5 = 567

699 - 6 = 693

Now find the greatest common factor of those 3 numbers:

441 = 3 x 3 x 7 x 7

572 = 3 x 3 x 3 x 3 x 7

693 = 3 x 3 x 7 x 11

The common factors are 3 x 3 x 7 = 63

HCF Of (441, 567, 693) = 63

445 / 63 = 7 remainder 4

572 / 63 = 9 remainder 5

699 / 63 = 11 remainder 6

∴ 63 is the largest divisor that will give the desired remainders.

QUESTION: 16

The product of a non-zero rational and irrational number is:

Solution:
Product of a non-zero rational number and an irrational number is always irrational.

QUESTION: 17

The HCF of the smallest composite number and the smallest prime number is:

Solution:
HCF of the smallest composite number and the smallest prime number = 2

Smallest composite number = 4 = 2 × 2

Smallest prime number = 2

QUESTION: 18

If HCF (306, 657) = 9, then LCM (306, 657) is;

Solution:
Given that HCF = 9 and the numbers are 306 and 657.

LCM = ?

We know that LCM × HCF = product of two numbers

⇒ LCM × 9 = 306 × 657

⇒ LCM = 201042 / 9 = 22338

QUESTION: 19

Write the exponent of 2 in the prime factorisation of 288.

Solution:
288 = 2 × 2 × 2 × 2 × 2 × 3 × 3

⇒ 2^{5} × 3²

Hence, The exponent of 2 is 5 and exponent of 3 is 2.

QUESTION: 20

What is the sum of exponents of prime factors in the prime factorisation of 98.

Solution:
98 = 2 × 7 × 7 = 2^{1} × 7^{2}

Sum of exponents = 1 + 2 = 3

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