Problems on HCF and LCM

Question 41

Question:   Find the HCF and LCM of 1152 and 1664 by using the fundamental theorem of arithmetic. [2 Marks]

Answer:    Steps:
1. Find the prime factors of 1152 and 1664
2. Find the common factors and write their products as HCF.
3. Find LCM by dividing the product of the 2 numbers by their HCF.

Prime factors of 1152 = 2⁷ x 3²
Prime factors of 1664 = 2⁷ x 13
HCF (1152, 1664) = 2⁷ = 128
LCM (1152, 1664) =
= 1152 x 13
= 14976
Therefore, HCF (1152, 1664) =128
LCM (1152, 1664) = 149

Question 42

Question:   Check whether 12ⁿ can end with the digit '0' for any (n belongs to N means n is a natural number).

Answer:    If the number 12ⁿ ends with '0', then it should be divisible by a prime number 5, which means that '5' is a prime factor of 12ⁿ.
This is not possible because, the prime factors of 12 are only 2 and 3. (12ⁿ) = (2 x 2 x 3)ⁿ. Here we do not have 5 as a factor. Hence by the principle of fundamental theorem of arithmetic, we have no other prime in the factorization of 12ⁿ other than 2 and 3. So we can say that there is no natural number n for which 12ⁿ ends with the digit zero.

Question 43

Question:   Find why 5 x 11 x 19 x 23 + 23 is a composite number.

Answer:    23 is a common factor. So the given number can be written as
23 (5 x 11 x 19) + 23
= 23 (1045) + 23
= 23 (1045 +1)
= 23 x 1046 = 2 x 23 x 528
This is the product of prime numbers. Therefore the given number is a composite number.

Question 44

Question:   Prove that is an irrational number by contradiction method. [3 Marks]

Answer:    Let assume that is a rational number.
Write (p and q are co-prime, they are integers and )

Squaring both sides

p² = 3q² or 3q² = p²
==> 3 divides p² and 3 divides p …(1)
Let p = 3x
p² = 9x²
So, 3q² = 9x²
q² = 3x²
==> 3 divides q² and 3 divides q …(2)
From (1) and (2), we get 3 is a common factor of p and q. This contradicts the assumption that p and q are co-prime.
Therefore is not a rational number.
Hence is an irrational numbe

Question 45

Question:   Prove that is an irrational number by representing it in decimal form. [2 Marks]

Answer:    Prove that is irrational
is a non-terminating and non-repeating decimal.
is irrational.

Question 46

Question:   Prove that is an irrational number by contradiction method. [3 Marks]

Answer:    Prove that is irrational by contradiction method.
Let us assume that is rational number.
(p and q are integers, q 0, p and q have no common factor)
Squaring both the sides




Let p = 5y ==> p² = 25y² 5q² = 25y², q² = 5y²
==> 5 divides q²
==> 5 divides q
From (1) and (2) we get 5 is a common factor of p and q.
This contradicts the assumption that p and q have no common factor.
is an irrational number

Question 47

Question:   Prove that is an irrational number by contradiction method. [3 Marks]

Answer:    Prove that is irrational by contradiction method.
Let us assume that is a rational number.
(p and q are integers, p and q are co-prime)
Squaring both the sides,

p² = 7q² 7 divides p²and 7 divides p …(1)
Let p = 7x
4.9x² = 7q² 7x² = q²
7 divides q²
7 divides q …(2)
From (1) and (2) we get '7' is a common factor of p and q, which is contradicting our assumption that p and q are co-prime.
is an irrational number.

Question 48

Question:   Show that is an irrational number by contradiction method. [2 Marks]

Answer:    Show that is an irrational number.
Let us assume that a rational number

Let
Here y is rational, 3 is rational
is rational which implies is rational. But we know that is irrational. So our assumption is wrong.
is an irrational number.

Question 49

Question:   Without actual division, say whether the following are terminating decimal or non terminating repeating decimals.
(i), (ii) , (iii) , (iv) , (v)

Answer:    (i) . Here factors of 80 are in the form 2⁴ x 5. (2ⁿ 5^m). So it is terminating decimal.
(ii) which when simplifies becomes which is a terminating decimal.
(iii) ; factors of 250 = 2 x 5³ …. Terminating decimal
(iv) ; factors of 17, 1 and 17 are non terminating repeating decimal
(v) ; factors of 15 = 3 x 5: '3' is a factor, so it is non terminating repeating decimal.