Finding Common Ground | FIO

Question 8

The LCM of 3 and 24 is 24 (it is one of the two given numbers).

(a) Find more such number pairs where the LCM is one of the two numbers.

(b) Make a general statement about such numbers. Describe such number pairs using algebra.

Solution

The Least Common Multiple (LCM) of two numbers can be one of the numbers.

Step 1 — Understanding the example

We are given an example pair of numbers. The numbers are 3 and 24. Their LCM is 24. Notice that 3 is a factor of 24. This means 24 can be divided by 3 exactly. The LCM is the larger number in this pair.

Step 2 — Finding more number pairs

Let us find more pairs like this. We need a smaller number that divides a larger number. The LCM will then be the larger number.

Let us pick 2 as the smaller number. We need a multiple of 2. Let us pick 4. Multiples of 2 are: 2, 4, 6, 8, ... Multiples of 4 are: 4, 8, 12, ... The LCM of 2 and 4 is 4. This pair works.

Let us pick 5 as the smaller number. We need a multiple of 5. Let us pick 10. Multiples of 5 are: 5, 10, 15, 20, ... Multiples of 10 are: 10, 20, 30, ... The LCM of 5 and 10 is 10. This pair works.

Let us pick 6 as the smaller number. We need a multiple of 6. Let us pick 12. The LCM of 6 and 12 is 12. This pair works.

Let us pick 7 as the smaller number. We need a multiple of 7. Let us pick 49. The LCM of 7 and 49 is 49. This pair works.

Let us pick 10 as the smaller number. We need a multiple of 10. Let us pick 100. The LCM of 10 and 100 is 100. This pair works.

Step 3 — Making a general statement

We have seen a pattern in these pairs. The smaller number is always a factor of the larger number. When this happens, the LCM is the larger number. So, the LCM of two numbers is the larger number. This is true if the smaller number divides the larger number. It is true only if the smaller number divides the larger number.

Step 4 — Describing with algebra

Let us use letters for the numbers. Let the two positive integers be $a$ and $b$ . Let us assume $a$ is the smaller number. So, $a < b$ . We want the LCM of $a$ and $b$ to be $b$ . This happens if $b$ is a multiple of $a$ . This means $b$ can be divided by $a$ exactly. So, $b$ is $a$ multiplied by some whole number. Let us call this whole number $k$ . $k$ must be a positive integer. So, we can write the relationship as: $b = k \times a$

Answer

(a) Examples of such pairs: (i) 2 and 4: LCM(2, 4) = 4 (ii) 5 and 10: LCM(5, 10) = 10 (iii) 6 and 12: LCM(6, 12) = 12 (iv) 7 and 49: LCM(7, 49) = 49 (v) 10 and 100: LCM(10, 100) = 100 (b) General Statement: For any two positive integers, the LCM is equal to the larger number if and only if the smaller number is a factor of the larger number. Algebraic Description: Let the two positive integers be $a$ and $b$ , where $a < b$ . The LCM of $a$ and $b$ is $b$ if and only if $b = k \times a$ , where $k$ is a positive integer.

More questions in FIO

List all the factors of the following numbers:

(a) 90

(b) 105

(d) 360 (this number has 24 factors)

(e) 840 (this number has 32 factors)

Find the common factors and the HCF of the following numbers:

(a) 50, 60

(b) 140, 275

(d) 370, 592

(e) 81, 243

How do we directly find the HCF without listing all the factors?

Find the HCF of the following numbers:

(a) 24, 180

(b) 42, 75, 24

(d) 400, 2500

(e) 300, 800

Consider the numbers 72 and 144. Suppose they are factorised into composite numbers as: 72 = 6 × 12 and 144 = 8 × 18. Seeing this, can one say that these two numbers have no common factor other than 1? Why not?

Find the LCM of the following numbers:

(a) 30, 72

(b) 36, 54

(d) 222, 370

Make a general statement about the HCF for the following pairs of numbers. You could consider examples before coming up with general statements. Look for possible explanations of why they hold.

(a) Two consecutive even numbers

(b) Two consecutive odd numbers

(d) Two consecutive numbers

(e) Two co-prime numbers

Share your observations with the class.

The LCM of 3 and 24 is 24 (it is one of the two given numbers).

(a) Find more such number pairs where the LCM is one of the two numbers.

(b) Make a general statement about such numbers. Describe such number pairs using algebra.

Make a general statement about the LCM for the following pairs of numbers. You could consider examples before coming up with these general statements. Look for possible explanations of why they hold.

(a) Two multiples of 3

(b) Two consecutive even numbers

(d) Two co-prime numbers

Q10

In the two rows below, colours repeat as shown. When will the blue stars meet next?

Q11

(a) Is $5 \times 7 \times 11 \times 11$ a multiple of $5 \times 7 \times 7 \times 11 \times 2$ ?

(b) Is $5 \times 7 \times 11 \times 11$ a factor of $5 \times 7 \times 7 \times 11 \times 2$ ?

Q12

Find the HCF and LCM of the following (state your answers in the form of prime factorisations):

(a) $3 \times 3 \times 5 \times 7 \times 7$ and $12 \times 7 \times 11$

(b) 45 and 36

Q13

Find two numbers whose HCF is 1 and LCM is 66.

Q14

A cowherd took all his cows to graze in the fields. The cows came to a crossing with 3 gates. An equal number of cows passed through each gate. Later at another crossing with 5 gates again an equal number of cows passed through each gate. The same happened at the third crossing with 7 gates. If the cowherd had less than 200 cows, how many cows did he have? (Based on the folklore mathematics from Karnataka.)

Q15

The length, width, and height of a box are 12 cm, 18 cm, and 36 cm respectively. Which of the following sized cubes can be packed in this box without leaving gaps?

(a) 9 cm

(b) 6 cm

(d) 3 cm

(e) 2 cm

Q16

Among the numbers below, which is the largest number that perfectly divides both 306 and 36?

(a) 36

(b) 612

(d) 3

(e) 2

(f) 360

Q17

Find the smallest number that is divisible by 3, 4, 5 and 7, but leaves a remainder of 10 when divided by 11.

Q18

Children are playing ‘Fire in the Mountain’. When the number 6 was called out, no one got out. When the number 9 was called out, no one got out. But when the number 10 was called out, some people got out. How many children could have been playing initially?

(a) 72

(b) 90

(d) 3

(e) 36

(f) None of these

Q19

Tick the correct statement(s). The LCM of two different prime numbers ( $m, n$ ) can be:

(a) Less than both numbers

(b) In between the two numbers

(d) Less than $m \times n$

(e) Greater than $m \times n$

Q20

A dog is chasing a rabbit that has a head start of 150 feet. It jumps 9 feet every time the rabbit jumps 7 feet. In how many leaps does the dog catch up with the rabbit?

Q21

What is the smallest number that is a multiple of 1, 2, 3, 4, 5, 6, 8, 9, 10? Do you remember the answer from Grade 6, Chapter 5?

Q22

Here is a problem posed by the ancient Indian Mathematician Mahaviracharya (850 C.E.). Add together $\frac{8}{15}$ , $\frac{1}{20}$ , $\frac{7}{36}$ , $\frac{11}{63}$ and $\frac{1}{21}$ . What do you get? How can we find this sum efficiently?