Exam Questions
3) Jasmine says “there are no prime numbers between 100 and 110.” Is she correct? Give evidence for your answer.
To determine if Jasmine is correct, we can list the numbers from 101 to 109 and check which are prime. The prime numbers in this range are 101, 103, 107, and 109, demonstrating that Jasmine's claim is incorrect.
4) Mei thinks of a prime number. The sum of its digits is one more than a square number. Write down one number Mei could be thinking of.
One possible prime number she could be thinking of is 11, as the sum of its digits (1 + 1) equals 2, which is one more than the square number 1 (i.e., 1² = 1).
5) Write 72 as a product of its prime factors.
To do this, we start by dividing 72 by the smallest prime number, which is 2: 72 ÷ 2 = 36. Continuing to divide by 2, we get 36 ÷ 2 = 18, and then 18 ÷ 2 = 9. Since 9 is not divisible by 2, we move to the next prime number, which is 3: 9 ÷ 3 = 3, and finally 3 ÷ 3 = 1. Therefore, the prime factorization of 72 is 2 × 2 × 2 × 3 × 3, or in exponential form, it can be expressed as 2² × 3².
6) P = 3^7 × 11² and Q = 3^4 × 7³ x 11.
Write as the product of prime factors:
a) the LCM of P and Q. To find the LCM, we take the highest power of each prime factor present in either number:
For 3, the highest power is 3^7.
For 7, the highest power is 7^3.
For 11, the highest power is 11².
Thus, the LCM of P and Q is 3^7 × 7^3 × 11².
b) the HCF of P and Q.
The HCF of P and Q is determined by taking the lowest power of each prime factor present in both numbers. Hence, we have:
For 3, the lowest power is 3^0 (since it does not appear in Q)
For 7, the lowest power is 7^3
For 11, the lowest power is 11^0 (since it does not appear in P)
Thus, the HCF of P and Q is 7³. To find the LCM, we take the highest powers of all prime factors: LCM = 3^1 * 7^3 * 11^1.