You are given that 270 = 3³ × 2 × 5 and 177 147 = 3¹¹ (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147. Give your answer using power notation and as an ordinary number. (ii) Write 177 147 000 000 as a product of its prime factors. (b) 3ⁿ = 177 147 × 9⁵. Find the value of n.

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You are given that 270 = 3³ × 2 × 5 and 177 147 = 3¹¹ (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147 - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Question 11

You are given that 270 = 3³ × 2 × 5 and 177 147 = 3¹¹ (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147. Give your answer using power notation and a... show full transcript

Worked Solution & Example Answer:You are given that 270 = 3³ × 2 × 5 and 177 147 = 3¹¹ (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147 - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Step 1

(i) Find the lowest common multiple (LCM) of 270 and 177 147.

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Answer

To find the LCM using the prime factorizations:

The prime factorization of 270 is: $3^3 \times 2^1 \times 5^1$ .
The prime factorization of 177 147 is: $3^{11}$ .

To determine the LCM, take the highest power of each prime:

For $3$ : the highest power is $3^{11}$ .
For $2$ : the highest power is $2^1$ (from 270).
For $5$ : the highest power is $5^1$ (from 270).

Thus, the LCM in power notation is: $LCM = 3^{11} \times 2^1 \times 5^1$

In ordinary number form:

$LCM = 3^{11} \times 2 \times 5 = 1771470$ .

Step 2

(ii) Write 177 147 000 000 as a product of its prime factors.

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Answer

Given the number: 177 147 000 000, we can express it as follows:

First, recognize that this number can be rewritten with prime factors: $177 147 000 000 = 177 147 imes 10^{6}$

Breaking down $10^{6}$ , we have: $10^6 = (2^1 imes 5^1)^{6} = 2^6 imes 5^6$

Next, using the prime factorization for 177 147, $177 147 = 3^{11}$

Thus, we combine these together:

$177 147 000 000 = 3^{11} imes 2^6 imes 5^6$

Step 3

Find the value of n.

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Answer

From the equation: $3^n = 177147 \times 9^5$

First, note that $9^5 = (3^2)^5 = 3^{10}$ .

Thus, we have: $3^n = 177147 \times 3^{10}$

Recalling that $177 147 = 3^{11}$ , we rewrite it: $3^n = 3^{11} \times 3^{10}$

Combine the powers: $3^n = 3^{21}$

From the property of exponents, we equate exponents: $n = 21$

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Questions answered

You are given that 270 = 3³ × 2 × 5 and 177 147 = 3¹¹ (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147 - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Worked Solution & Example Answer:You are given that 270 = 3³ × 2 × 5 and 177 147 = 3¹¹ (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147 - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

(i) Find the lowest common multiple (LCM) of 270 and 177 147.

(ii) Write 177 147 000 000 as a product of its prime factors.

Find the value of n.

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