From the list of numbers, write down61 62 63 64 65 66 67 68 69a cube number.

Goal: Identify the cube number from the given list of numbers.Step 1: A cube number is an integer that is the result of cubing another integer (i.e., raising it to the power of 3). We need to find which number in the list is a perfect cube.Step 2: We can test the cubes of small integers to see if they match any number in the list.The number is too small and is too large. The number is the cube of and it is in the given list. M1Conclusion: The cube number from the list is 64.

61 63 64 66 68 69From this list, write down a cube number.

Goal: Identify the cube number from the list 61, 63, 64, 66, 68, 69.Step 1: A cube number is a number that is the result of multiplying an integer by itself three times. We need to check which number in the given list is a perfect cube.Step 2: We can test the cubes of small integers to see if any of them match a number in the list. Let's calculate the cube of 4:The number 64 is in the list {61, 63, 64, 66, 68, 69}.Conclusion: The cube number from the list is 64.

is an integer. is a cube number.Find the smallest possible value of .

Goal: Find the smallest possible value of `R` such that is a cube number.Step 1: Analyze the prime factorization of and the condition for a cube number.We are given .For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.The expression must therefore have prime factors with exponents that are multiples of 3.Step 2: Determine the factors needed for to satisfy the condition.To find the smallest positive integer , we need to divide by the necessary factors to reduce the exponents to the nearest multiple of 3 (that is less than or equal to the original exponent).For the prime factor 2, the exponent is 7. The largest multiple of 3 less than or equal to 7 is 6. To change the exponent from 7 to 6, we must divide by .For the prime factor 3, the exponent is 3. This is already a multiple of 3, so we don't need to divide by any power of 3 (or we divide by ).For the prime factor 5, the exponent is 2. The largest multiple of 3 less than or equal to 2 is 0. To change the exponent from 2 to 0, we must divide by .These factors make up the value of . So, .Step 3: Calculate the value of .With this value of , , which is a perfect cube.Conclusion: The smallest possible positive integer value of is 50.

120 121 149 164 216From this list, write down a cube number.

Goal: Identify the cube number from the list of numbers provided.Step 1: A cube number is a number that is the result of multiplying an integer by itself three times (i.e., ). We will test the numbers in the list to see which one is a perfect cube.Step 2: Let's calculate the first few positive integer cubes to see if any match the numbers in the list: 120, 121, 149, 164, 216.By comparing our calculated cube numbers with the given list, we can see that 216 is a cube number.Conclusion: The cube number from the list is 216.

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IGCSE (CAMBRIDGE)|9th and 10th|Extended Maths (0580)

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