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Andrew is thinking of a number - OCR - GCSE Maths - Question 3 - 2017 - Paper 1
Question 3
Andrew is thinking of a number. - It is between 1 and 150. - It is one more than a square number. - It is three less than a cube number. - It is not a prime number.... show full transcript
Worked Solution & Example Answer:Andrew is thinking of a number - OCR - GCSE Maths - Question 3 - 2017 - Paper 1
Step 1
It is one more than a square number.
Answer
To find a number that is one more than a square number, we can express it as:
for integers k. The square numbers less than 150 are 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Thus, possible values for n are:
- 1 (for k=0)
- 2 (for k=1)
- 5 (for k=2)
- 10 (for k=3)
- 17 (for k=4)
- 26 (for k=5)
- 37 (for k=6)
- 50 (for k=7)
- 65 (for k=8)
- 82 (for k=9)
- 101 (for k=10)
- 122 (for k=11)
- 145 (for k=12)
Step 2
It is three less than a cube number.
Answer
The condition that the number is three less than a cube number can be represented as:
for integers m. The cube numbers less than 150 are 0, 1, 8, 27, 64, 125. Thus, possible values for n are:
- -3 (for m=0)
- -2 (for m=1)
- 5 (for m=2)
- 24 (for m=3)
- 61 (for m=4)
- 122 (for m=5)
Step 3
It is not a prime number.
Answer
From the previous steps, we found potential values for n that are:
- From squares: {1, 2, 5, 10, 17, 26, 37, 50, 65, 82, 101, 122, 145}
- From cubes: {5, 24, 61, 122}
The intersection gives us: {5, 122}.
We need to exclude any prime numbers from these. The only candidates are:
- 5 (which is prime)
- 122 (which is not prime)
Thus, the only valid number that satisfies all conditions is 122.