You are given that 270 = 3^3 × 2 × 5 and 177 147 = 3^{11}. (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) 3^n = 177 147 × 9^5. Find the value of n.

GCSE OCR Maths Indices: You are given that 270 = 3^3 ×

You are given that 270 = 3^3 × 2 × 5 and 177 147 = 3^{11} - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Question 11

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You are given that 270 = 3^3 × 2 × 5 and 177 147 = 3^{11}. (a) (i) Find the lowest common multiple (LCM) of 270 and 177 147. Give your answer using power notation ... show full transcript

Worked Solution & Example Answer:You are given that 270 = 3^3 × 2 × 5 and 177 147 = 3^{11} - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Step 1

Find the lowest common multiple (LCM) of 270 and 177 147 using power notation

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Answer

The prime factorization of 270 is given as:

$270 = 3^3 × 2^1 × 5^1.$

The prime factorization of 177 147 is:

$177147 = 3^{11}.$

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

For 3: maximum of $3^3$ and $3^{11}$ is $3^{11}$ .
For 2: only present in 270 as $2^1$ .
For 5: only present in 270 as $5^1$ .

Thus,

$LCM = 3^{11} × 2^1 × 5^1.$

In ordinary number notation, this is:

$3^{11} × 2 × 5 = 177 147 × 10 = 1 771 470.$

Step 2

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

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Answer

To express 177 147 000 000 as a product of its prime factors, start with the factorization of 177 147:

$177 147 = 3^{11}.$

Now, take into account that 177 147 000 000 can be expressed as:

$177 147 000 000 = 177 147 × 10^{6} = 3^{11} × (2^1 × 5^1)^{6} = 3^{11} × 2^{6} × 5^{6}.$

Step 3

Find the value of n in 3^n = 177 147 × 9^5.

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Answer

Transform 9 to its prime factors:

$9 = 3^2.$

Thus,

$9^5 = (3^2)^5 = 3^{10}.$

Substituting this back, we have:

$3^n = 177 147 × 9^5 = 3^{11} × 3^{10} = 3^{21}.$

Therefore, by equating the powers of 3, we find:

$n = 21.$

You are given that 270 = 3^3 × 2 × 5 and 177 147 = 3^{11} - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Worked Solution & Example Answer:You are given that 270 = 3^3 × 2 × 5 and 177 147 = 3^{11} - OCR - GCSE Maths - Question 11 - 2019 - Paper 6

Find the lowest common multiple (LCM) of 270 and 177 147 using power notation

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

Find the value of n in 3^n = 177 147 × 9^5.

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