Mathematics
State whether the following statements are true (T) or false (F):
(i) Every natural number has a finite number of factors.
(ii) Every natural number has an infinite number of its multiples.
(iii) There are infinitely many prime numbers.
(iv) The HCF of two numbers is smaller than the smaller of the numbers.
(v) The LCM of two numbers is greater than the larger of the numbers.
Answer
(i) True.
Reason: Every natural number has a limited number of factors, with the largest factor being the number itself.
(ii) True.
Reason: Every natural number can be multiplied by infinitely many natural numbers giving infinite multiples.
(iii) True.
Reason: It has been proven (since the time of Euclid) that there are infinitely many prime numbers.
(iv) False.
Reason: The HCF of two numbers is less than or equal to the smaller of the two numbers. For example, HCF(6, 12) = 6, which is equal to the smaller number.
(v) False.
Reason: The LCM of two numbers is greater than or equal to the larger of the two numbers. For example, LCM(6, 12) = 12, which is equal to the larger number.
Related Questions
Can two numbers have 15 as their HCF and 110 as their LCM? Give reason to justify your answer.
Fill in the blanks:
(i) The only natural number which has exactly one factor is ….
(ii) The only prime number which is even is ….
(iii) The HCF of two co-prime numbers is ….
(iv) Two perfect numbers are … and ….
(v) The only prime-triplet is ….
(vi) The LCM of two or more given numbers is the lowest their common …
(vii) The HCF of two or more of given numbers is the highest of their common …
State whether the following statements are true or false. If a statement is false, justify your answer:
(i) The sum of two prime numbers is always an even number.
(ii) The sum of two prime numbers is always a prime number.
(iii) The sum of two prime numbers can never be a prime number.
(iv) No odd number can be written as the sum of two prime numbers.
(v) If two numbers are co-prime, then atleast one of them must be prime.
(vi) If a number is divisible by 18, it must be divisible by 3 and 6 both.
(vii) If a number is divisible by 2 and 4 both, it must be divisible by 8.
(viii) If a number is divisible by 3 and 6 both, it must be divisible by 18.
(ix) HCF of an even number and an odd number is always 1.
All factors of 6 are
1, 6
2, 3
1, 2, 3
1, 2, 3, 6