(a) From the list of numbers: 6, 7, 10, 12, 18, 49, 63, write down:
(i) a factor of 21
(ii) a square number
(iii) a prime number
(b) Find the value of:
(i) the cube root of 1728
(ii) \(2^5\)
(iii) \(5^0\)
(iv) \(36^{\frac{1}{2}}\)
(c) Put one pair of brackets into this calculation to make it correct:
$$3 \times 2 – 6 – 2 \div 2 = 4$$
(d) Find the lowest common multiple (LCM) of 30 and 68.
▶️ Answer/Explanation
(a)(i) Ans: 7
Factors of 21: 1, 3, 7, 21. From the list, only 7 is present.
(a)(ii) Ans: 49
Square numbers in list: 49 (since \(7 \times 7 = 49\)).
(a)(iii) Ans: 7
Prime numbers in list: 7 (only divisible by 1 and itself).
(b)(i) Ans: 12
\(\sqrt[3]{1728} = 12\) because \(12 \times 12 \times 12 = 1728\).
(b)(ii) Ans: 32
\(2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32\).
(b)(iii) Ans: 1
Any non-zero number to the power of 0 is 1: \(5^0 = 1\).
(b)(iv) Ans: 6
\(36^{\frac{1}{2}} = \sqrt{36} = 6\).
(c) Ans: \(3 \times 2 – (6 – 2) \div 2 = 4\)
Working: \(3 \times 2 – (6 – 2) \div 2 = 6 – 4 \div 2 = 6 – 2 = 4\)
(d) Ans: 1020
Prime factors: \(30 = 2 \times 3 \times 5\) \(68 = 2^2 \times 17\) LCM = \(2^2 \times 3 \times 5 \times 17 = 1020\)
(a)
(i) 1 and 120 are factors of 120. Write down another factor of 120.
(ii) Find the highest common factor of 120 and 900.
(b) From the list: 2, 5, 15, 24, 49, 60, 258, 512, write down:
(i) a multiple of 30,
(ii) a square number,
(iii) the cube root of 8.
(c) Give an example to show that the following statements are not true:
(i) An odd number multiplied by an even number gives an odd number.
(ii) The cube of a negative number is positive.
(d) Use < , > , or = to complete the following statements:
(i) 0.5 __ \(\frac{3}{8}\)
(ii) 1.5 __105%
(iii) 0.78 __\(\frac{11}{14}\)
▶️ Answer/Explanation
(a)(i) Ans: 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, or 60
120 is divisible by these numbers (e.g., 120 ÷ 2 = 60).
(a)(ii) Ans: 60
Prime factors of 120: 2³ × 3 × 5
Prime factors of 900: 2² × 3² × 5²
HCF: Take lowest powers → 2² × 3 × 5 = 60
(b)(i) Ans: 60
60 is a multiple of 30 (30 × 2 = 60).
(b)(ii) Ans: 49
49 is a square number (7 × 7 = 49).
(b)(iii) Ans: 2
2 is the cube root of 8 (2³ = 8).
(c)(i) Ans: Example: 3 × 2 = 6 (even result)
Odd × Even = Even, disproving the statement.
(c)(ii) Ans: Example: (-2)³ = -8 (negative result)
Cubes of negative numbers are negative.
(d)(i) Ans: >
0.5 = 0.5 > \(\frac{3}{8}\) ≈ 0.375
(d)(ii) Ans: >
1.5 > 105% (which is 1.05)
(d)(iii) Ans: <
0.78 < \(\frac{11}{14}\) ≈ 0.7857