Page 64 - Start Up Mathematics_6

Solution: (a) The HCF of two consecutive numbers is 1, as 1 is the only common factor in
two consecutive numbers.
(b) The HCF of two consecutive odd numbers is 1, as 1 is the only common factor
in two consecutive odd numbers.
(c) The HCF of two consecutive even numbers is 2, as 2 is the only common factor
in two consecutive even numbers.
(d) The HCF of any two primes hence two consecutive prime numbers is 1.
Example 38: The HCF of co-prime numbers 9 and 14 was worked out as follows by factorization:
9 = 3 × 3 and 14 = 2 × 7. Since, there is no common prime factor, so the HCF of 9
and 14 is 0. Is the answer correct? If not, what is the correct HCF?
Solution: No; the answer is not correct.
Prime factors of 9 = 3 × 3 and prime factors of 14 = 2 × 7.

Apparently, there is no common prime factor in 9 and 14 but 1 is always a factor of
any number. Therefore, the HCF of 9 and 14 is 1 and not 0.

Lowest Common Multiple (LCM)

The lowest common multiple (LCM) of two or more numbers is the lowest or smallest of their
common multiples. For example, to find LCM of 8 and 12 we write their common multiples 24,
48, 72, 96, .... The lowest of these is 24. Hence, LCM of 8 and 12 is 24.

LCM by prime factorization

Step 1: Find prime factors of each of the given numbers.
Step 2: Identify the maximum number of times each prime factor appears.
Step 3: The product of these prime factors with greatest powers is the LCM.
Example 39: Find the LCM of 8 and 12.

Solution: Prime factors of 8 = 2 × 2 × 2 ; Prime factors of 12 = 2 × 2 × 3
In these prime factorizations, the maximum number of times the prime factor 2 occurs
is three; i.e., 2 × 2 × 2 = 8. Similarly, the maximum number of times the factor 3
occurs is one; i.e., 3.
Therefore, the LCM of 8 and 12 = (2 × 2 × 2) × 3 = 8 × 3 = 24.

LCM by division method

Step 1: Write the given numbers in a row separated by commas in any order.
Step 2: Find the least prime number which divides at least two of the given numbers exactly.
Step 3: Write the quotients just below the respective numbers. The numbers which are not
divisible by the least prime number are written as it is below the respective numbers.
Step 4: Keep on repeating Step 2 till no two numbers are divisible by the same number.
Step 5: To get the LCM, find the product of the divisors and the remaining quotients.

Example 40: Find the LCM of 15, 30 and 90. 2 15, 30, 90
Solution: Therefore, LCM = 2 × 3 × 5 × 3 = 90. 3 15, 15, 45
5 5, 5, 15
1, 1, 3

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