Finding Common Ground | IT

Question 26

Explore whether the LCM is a factor of the product in the following cases. If yes, identify the number that the LCM should be multiplied by to get the product. Do you see any pattern? Use these numbers:

(a) 45, 105

(b) 275, 352

Solution

We explore the relationship between the product and LCM of two numbers.

Step 1 — Analyzing 45 and 105

Let us find the prime factors of each number. This helps us find the LCM easily.

The number 45 is $3 \times 15$ . It is $3 \times 3 \times 5$ . So, $45 = 3^2 \times 5^1$ .

The number 105 is $3 \times 35$ . It is $3 \times 5 \times 7$ . So, $105 = 3^1 \times 5^1 \times 7^1$ .

Now, we find the product of these numbers. Let P be the product.

$P = 45 \times 105$

$P = 4725$

$\boxed{\text{Product} = \mathbf{4725}}$

<DIAGRAM: Two numbers, 45 and 105, with their prime factorizations shown as factor trees. 45 -> 3, 15 -> 3, 5. 105 -> 3, 35 -> 5, 7. A multiplication sign between 45 and 105, leading to 4725.>

Step 2 — LCM and Multiplier for 45, 105

We find the LCM of 45 and 105. We take the highest power of each prime factor. The prime factors are 3, 5, and 7. Highest power of 3 is $3^2$ . Highest power of 5 is $5^1$ . Highest power of 7 is $7^1$ .

$LCM(45, 105) = 3^2 \times 5^1 \times 7^1$

$= 9 \times 5 \times 7$

$= 45 \times 7$

$= 315$

$\boxed{LCM(45, 105) = \mathbf{315}}$

Now, we check if LCM is a factor of the product. We divide the product by the LCM.

$\text{Multiplier} = \frac{\text{Product}}{\text{LCM}}$

$= \frac{4725}{315}$

$= 15$

$\boxed{\text{Multiplier} = \mathbf{15}}$

Yes, the LCM is a factor of the product. The multiplier is 15. Let us find the HCF of 45 and 105. We take the lowest power of common prime factors. Common prime factors are 3 and 5. Lowest power of 3 is $3^1$ . Lowest power of 5 is $5^1$ .

$HCF(45, 105) = 3^1 \times 5^1$

$= 3 \times 5$

$= 15$

$\boxed{HCF(45, 105) = \mathbf{15}}$

The multiplier is equal to the HCF.

Step 3 — Analyzing 275 and 352

Let us find the prime factors of each number.

The number 275 is $5 \times 55$ . It is $5 \times 5 \times 11$ . So, $275 = 5^2 \times 11^1$ .

The number 352 is $2 \times 176$ . It is $2 \times 2 \times 88$ . It is $2 \times 2 \times 2 \times 44$ . It is $2 \times 2 \times 2 \times 2 \times 22$ . It is $2 \times 2 \times 2 \times 2 \times 2 \times 11$ . So, $352 = 2^5 \times 11^1$ .

Now, we find the product of these numbers. Let P be the product.

$P = 275 \times 352$

$P = 96800$

$\boxed{\text{Product} = \mathbf{96800}}$

<DIAGRAM: Two numbers, 275 and 352, with their prime factorizations shown as factor trees. 275 -> 5, 55 -> 5, 11. 352 -> 2, 176 -> 2, 88 -> 2, 44 -> 2, 22 -> 2, 11. A multiplication sign between 275 and 352, leading to 96800.>

Step 4 — LCM and Multiplier for 275, 352

We find the LCM of 275 and 352. We take the highest power of each prime factor. The prime factors are 2, 5, and 11. Highest power of 2 is $2^5$ . Highest power of 5 is $5^2$ . Highest power of 11 is $11^1$ .

$LCM(275, 352) = 2^5 \times 5^2 \times 11^1$

$= 32 \times 25 \times 11$

$= 800 \times 11$

$= 8800$

$\boxed{LCM(275, 352) = \mathbf{8800}}$

Now, we check if LCM is a factor of the product. We divide the product by the LCM.

$\text{Multiplier} = \frac{\text{Product}}{\text{LCM}}$

$= \frac{96800}{8800}$

$= 11$

$\boxed{\text{Multiplier} = \mathbf{11}}$

Yes, the LCM is a factor of the product. The multiplier is 11. Let us find the HCF of 275 and 352. We take the lowest power of common prime factors. Common prime factor is 11. Lowest power of 11 is $11^1$ .

$HCF(275, 352) = 11^1$

$= 11$

$\boxed{HCF(275, 352) = \mathbf{11}}$

The multiplier is equal to the HCF.

Step 5 — Analyzing 222 and 370

Let us find the prime factors of each number.

The number 222 is $2 \times 111$ . It is $2 \times 3 \times 37$ . So, $222 = 2^1 \times 3^1 \times 37^1$ .

The number 370 is $10 \times 37$ . It is $2 \times 5 \times 37$ . So, $370 = 2^1 \times 5^1 \times 37^1$ .

Now, we find the product of these numbers. Let P be the product.

$P = 222 \times 370$

$P = 82140$

$\boxed{\text{Product} = \mathbf{82140}}$

<DIAGRAM: Two numbers, 222 and 370, with their prime factorizations shown as factor trees. 222 -> 2, 111 -> 3, 37. 370 -> 2, 185 -> 5, 37. A multiplication sign between 222 and 370, leading to 82140.>

Step 6 — LCM and Multiplier for 222, 370

We find the LCM of 222 and 370. We take the highest power of each prime factor. The prime factors are 2, 3, 5, and 37. Highest power of 2 is $2^1$ . Highest power of 3 is $3^1$ . Highest power of 5 is $5^1$ . Highest power of 37 is $37^1$ .

$LCM(222, 370) = 2^1 \times 3^1 \times 5^1 \times 37^1$

$= 2 \times 3 \times 5 \times 37$

$= 30 \times 37$

$= 1110$

$\boxed{LCM(222, 370) = \mathbf{1110}}$

Now, we check if LCM is a factor of the product. We divide the product by the LCM.

$\text{Multiplier} = \frac{\text{Product}}{\text{LCM}}$

$= \frac{82140}{1110}$

$= 74$

$\boxed{\text{Multiplier} = \mathbf{74}}$

Yes, the LCM is a factor of the product. The multiplier is 74. Let us find the HCF of 222 and 370. We take the lowest power of common prime factors. Common prime factors are 2 and 37. Lowest power of 2 is $2^1$ . Lowest power of 37 is $37^1$ .

$HCF(222, 370) = 2^1 \times 37^1$

$= 2 \times 37$

$= 74$

$\boxed{HCF(222, 370) = \mathbf{74}}$

The multiplier is equal to the HCF.

Step 7 — Discovering the Pattern

We have observed a consistent result. In all cases, the LCM is a factor of the product. The multiplier we found is special. It is always the HCF of the two numbers. So, Product = LCM $\times$ HCF. This is a very important property.

Answer

(a) Yes, the LCM is a factor of the product. The multiplier is 15. (b) Yes, the LCM is a factor of the product. The multiplier is 11. (c) Yes, the LCM is a factor of the product. The multiplier is 74. Pattern: The LCM is always a factor of the product. The multiplier is the HCF of the two numbers. So, for numbers A and B, A $\times$ B = LCM(A, B) $\times$ HCF(A, B).

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Context: Consider the number 840 and its prime factorisation $2 \times 2 \times 2 \times 3 \times 5 \times 7$ .

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Q11

Context: Consider the number 840 and its prime factorization $2 \times 2 \times 2 \times 3 \times 5 \times 7$ .

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Is $3 \times 3 \times 3$ a factor of 840? Why or why not?

Can we use this idea to list down all the possible factors of a number using just its prime factors?

Q12

Context: The factors of 225 are found to be 1, 3, 5, 9, 15, 25, 45, 75, 225.

Q. Check that all the factors of 225 occur in this list.

Q13

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Q14

Context: Consider the numbers 14 and 35, with prime factorisations $14 = 2 \times 7$ and $35 = 5 \times 7$ .

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Q15

Find more such number pairs where the HCF is one of the two numbers. How can we describe such pairs of numbers?

Q16

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Q17

What happens to the HCF of two numbers if both numbers are doubled? Take some pairs of numbers and explore. Are you able to see why the HCF will also double?

Q18

Here are some more numbers where both numbers are multiples of the same number. Find their HCF:

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(b) $10 \times 38$ , $10 \times 21$

(d) $12 \times 16$ , $12 \times 20$

Q19

In which of these cases is the HCF the same as the common multiplier, like problem (b) where the HCF is 10? Explore a few more examples of this type to understand when this happens.

Q20

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Q21

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Q22

Why are these the LCMs?

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Q23

You can try this method for these pairs of numbers.

(a) 90 and 150

(b) 84 and 132

Q24

Property Involving both the HCF and the LCM

Which is greater — the LCM of two numbers or their product?

Q25

You could analyse the above statement using examples. Then try to reason or prove, why the LCM is never greater than the product of the numbers.

[Hint: Is the product also a common multiple of the two numbers?]

Q26

(a) 45, 105

(b) 275, 352

Q27

Context: Consider the pairs of numbers: (a) 45, 105; (b) 275, 352; (c) 222, 370.

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Q28

Why does this happen? Can you give an explanation or proof?

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Q29

Explore whether this property holds when 3 numbers are considered.

Q30

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