3 (a) Write 504 as the product of its prime factors - OCR - GCSE Maths - Question 3 - 2017 - Paper 1

To find the LCM of 180 and 504, we first find the prime factorizations of both numbers:

For 180:

1. Divide 180 by 2: $180 \\ 2 = 90$
2. Divide 90 by 2: $90 \\ 2 = 45$
3. Divide 45 by 3: $45 \\ 3 = 15$
4. Divide 15 by 3: $15 \\ 3 = 5$
5. 5 is a prime number.

Thus, the prime factorization of 180 is: $180 = 2^2 \times 3^2 \times 5$

Now, using the factorizations:

• 504: $2^3 \times 3^2 \times 7$
• 180: $2^2 \times 3^2 \times 5$

The LCM is found by taking the highest powers of each prime:

• For 2: max(3, 2) = 3
• For 3: max(2, 2) = 2
• For 5: max(0, 1) = 1
• For 7: max(1, 0) = 1

Thus: $LCM(180, 504) = 2^3 \times 3^2 \times 5^1 \times 7^1$ Calculating this gives: $LCM(180, 504) = 8 \times 9 \times 5 \times 7 = 2520$