Number types and prime factors

1. Using the convention that natural numbers begin at 1, which number is a natural number?

Answer C: 3

The positive counting numbers are 1, 2, 3, …, so 3 is natural. The numbers −4 and 0 are integers but are not natural under the stated convention, and 1/2 is not an integer.

2. Which number is an integer but not a natural number under this pack's convention?

Answer A: −7

−7 is a negative whole number, so it is an integer. It is not a positive counting number, so it is not natural under the stated convention. The other options are not integers.

3. Which statement about 0 is correct?

Answer C: It is rational but not positive.

Zero can be written as 0/1, so it is rational. It is equal to 0, not greater than 0, so it is not positive. It is also not negative or irrational.

4. Which number is rational?

Answer C: 0.125

0.125 terminates and equals 125/1000 = 1/8, so it is rational. The numbers √3, π and √7 cannot be written as ratios of integers and are irrational.

5. Which number is irrational?

Answer D: √10

√10 is irrational because 10 is not a perfect square. By contrast, √49 = 7, −3/8 is already a ratio of integers, and 0.6 = 3/5, so those three are rational.

6. Which description includes both rational and irrational numbers?

Answer D: Real numbers

Real numbers consist of all rational and irrational numbers. Natural numbers, integers and prime numbers are narrower groups and contain no irrational numbers.

7. Which number is prime?

Answer C: 29

29 has only the positive factors 1 and 29, so it is prime. The number 1 has only one positive factor, 21 = 3 × 7, and 51 = 3 × 17.

8. Which number is a square number?

Answer B: 36

36 = 6², so it is a square number. None of 18, 54 or 72 is the square of an integer.

9. Which number is a cube number?

Answer A: −27

−27 = (−3)³, so it is a cube number. The other values are not cubes of integers.

10. Which number is a factor of 84?

Answer B: 7

84 ÷ 7 = 12 exactly, so 7 is a factor. Dividing 84 by 5, 9 or 11 leaves a remainder.

11. Which number is a common multiple of 6 and 8?

Answer B: 24

24 = 6 × 4 and 24 = 8 × 3, so it is a multiple of both numbers. Each other option fails at least one of the two divisibility checks.

12. What is the prime factorisation of 84?

Answer B: 2² × 3 × 7

84 = 2 × 42 = 2 × 2 × 21 = 2² × 3 × 7. Every final factor is prime. The expressions containing 14 or 4 are not fully prime-factorised, and 2 × 3² × 7 equals 126.

13. What is the HCF of 18 and 30?

Answer B: 6

The shared prime factors are 2 and one factor of 3, giving 2 × 3 = 6. Thus 6 divides both 18 and 30, and no larger common factor does.

14. What is the LCM of 12 and 15?

Answer C: 60

The LCM needs 2², 3 and 5, so it is 2² × 3 × 5 = 60. Indeed, 60 ÷ 12 = 5 and 60 ÷ 15 = 4, and no smaller positive common multiple exists.

15. What is the reciprocal of −3/5?

Answer B: −5/3

The reciprocal of −3/5 is −5/3. Their product is (−3/5) × (−5/3) = 1, which confirms the reciprocal.

16. What is the reciprocal of 0?

Answer D: It is undefined.

The reciprocal of x would be 1/x, but division by zero is undefined. Equivalently, 0 multiplied by any number is 0, never 1, so zero has no reciprocal.

17. What is the most specific classification of −3/5 among these choices?

Answer C: Rational

−3/5 is rational because it is a ratio of integers and the denominator is not zero. It is not an integer or natural number, and a rational number cannot also be irrational.

18. Which figure represents three billion, five million and forty-two?

Answer C: 3 005 000 042

The groups are 3 | 005 | 000 | 042: three billions, five millions, zero thousands and forty-two units. Therefore the figure is 3 005 000 042.

19. A number has prime factorisation 2³ × 3² × 5. Which value is a factor of the number?

Answer B: 2² × 3 × 5

2² × 3 × 5 uses no prime exponent beyond those in 2³ × 3² × 5, so it divides the number exactly. The other options require 2⁴, 3³ or 5², each exceeding an available exponent.

20. A floor measuring 168 cm by 252 cm is covered by the largest possible identical square tiles, with no cutting. What is the side length of each tile?

Answer C: 84 cm

This is an HCF problem. Since 168 = 2³ × 3 × 7 and 252 = 2² × 3² × 7, their HCF is 2² × 3 × 7 = 84. An 84 cm side divides both dimensions exactly, and no larger common factor exists.

21. Three signals flash every 18 seconds, 24 seconds and 30 seconds. They flash together now. After how many seconds will they next flash together?

Answer D: 360

Use LCM because all three repeating cycles must coincide. With 18 = 2 × 3², 24 = 2³ × 3 and 30 = 2 × 3 × 5, the LCM is 2³ × 3² × 5 = 360 seconds.

22. What is the HCF of 84 and 126?

Answer C: 42

For the HCF, take shared primes with the smaller exponents: 2¹ × 3¹ × 7¹ = 42. The checks 84 ÷ 42 = 2 and 126 ÷ 42 = 3 confirm that 42 divides both.

23. What is the LCM of 72 and 120?

Answer D: 360

For the LCM, take the greatest power of every prime present: 2³ × 3² × 5 = 8 × 9 × 5 = 360. Both 72 and 120 divide 360 exactly.

24. What is the smallest positive integer that is divisible by 8, 12 and 18?

Answer C: 72

The LCM takes 2³ and 3², giving 2³ × 3² = 8 × 9 = 72. It is divisible by 8, 12 and 18, so 72 is the smallest positive integer meeting all three conditions.