Chapter 3 — worked answers and marking guidance
All chapter review and practice examination items
Chapter review solutions
Find the 5th positive multiple of 9.
9 × 5 = 45.
Find the least positive common multiple of 7 and 13.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 91.
List all positive factors of 128.
Each factor divides 128 with no remainder; pair divisors with their matching quotients.
Classify 49 as prime, composite, or neither.
The positive factors are 1, 7, 49. Prime means exactly two positive factors; 1 is neither prime nor composite.
What is the greatest common factor of 18 and 39?
The GCF is the largest whole number dividing both with no remainder: 3.
Find the 2nd positive multiple of 12.
12 × 2 = 24.
Will 126 be a common multiple of 9 and 7? Explain.
126 is two times their least common multiple, so both numbers divide it.
List all positive factors of 20.
Each factor divides 20 with no remainder; pair divisors with their matching quotients.
Classify 71 as prime, composite, or neither.
The positive factors are 1, 71. Prime means exactly two positive factors; 1 is neither prime nor composite.
What is the greatest common factor of 20 and 33?
The GCF is the largest whole number dividing both with no remainder: 1.
Find the 3rd positive multiple of 10.
10 × 3 = 30.
Find the least positive common multiple of 6 and 12.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 12.
Is 3 a factor of 27? Give a reason.
27 ÷ 3 = 9, a whole number.
Classify 39 as prime, composite, or neither.
The positive factors are 1, 3, 13, 39. Prime means exactly two positive factors; 1 is neither prime nor composite.
What is the greatest common factor of 8 and 33?
The GCF is the largest whole number dividing both with no remainder: 1.
Find the 5th positive multiple of 15.
15 × 5 = 75.
Find the least positive common multiple of 8 and 7.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 56.
List all positive factors of 72.
Each factor divides 72 with no remainder; pair divisors with their matching quotients.
Classify 32 as prime, composite, or neither.
The positive factors are 1, 2, 4, 8, 16, 32. Prime means exactly two positive factors; 1 is neither prime nor composite.
What is the greatest common factor of 10 and 24?
The GCF is the largest whole number dividing both with no remainder: 2.
Find the 7th positive multiple of 8.
8 × 7 = 56.
Will 420 be a common multiple of 14 and 15? Explain.
420 is two times their least common multiple, so both numbers divide it.
List all positive factors of 64.
Each factor divides 64 with no remainder; pair divisors with their matching quotients.
Classify 77 as prime, composite, or neither.
The positive factors are 1, 7, 11, 77. Prime means exactly two positive factors; 1 is neither prime nor composite.
What is the greatest common factor of 8 and 45?
The GCF is the largest whole number dividing both with no remainder: 1.
Is zero a multiple of 16? Explain.
0 = 16 × 0, so zero is a multiple of 16.
Find the least positive common multiple of 5 and 5.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 5.
Is zero a multiple of 6? Explain using multiplication.
6 × 0 = 0. A multiple includes zero when the multiplier is a whole number.
A student says 48 is the least positive common multiple of 6 and 8 because both divide it. Find the actual least positive common multiple.
24 is the first positive multiple shared by 6 and 8; 48 is a later common multiple.
A learner says 12 has only factors 2, 3, and 4. Explain the omitted factors and list all positive factors.
Factor pairs of 12 are (1,12), (2,6), and (3,4).
Sam claims 13 is prime because it is not divisible by 5. Explain why that test is insufficient, and classify 13.
Not being divisible by 5 proves nothing by itself. The factors of 13 are 1, 13. A prime has exactly two positive factors.
A learner says the greatest common factor of 12 and 18 is 36 because both multiply into 36. Find the GCF.
A common factor divides both numbers; 6 is the largest positive divisor shared by 12 and 18.
Is zero a multiple of 9? Explain using multiplication.
9 × 0 = 0. A multiple includes zero when the multiplier is a whole number.
Practice examination solutions and scoring
Find the 8th positive multiple of 12.
12 × 8 = 96.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Is 15 a factor of 180? Give a reason.
180 ÷ 15 = 12, a whole number.
Scoring (2 points): Award one point for a sound method and one for the correct result.
What is the greatest common factor of 8 and 21?
The GCF is the largest whole number dividing both with no remainder: 1.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the least positive common multiple of 14 and 11.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 154.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Classify 59 as prime, composite, or neither.
The positive factors are 1, 59. Prime means exactly two positive factors; 1 is neither prime nor composite.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Will 420 be a common multiple of 15 and 14? Explain.
420 is two times their least common multiple, so both numbers divide it.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Classify 46 as prime, composite, or neither.
The positive factors are 1, 2, 23, 46. Prime means exactly two positive factors; 1 is neither prime nor composite.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the 5th positive multiple of 8.
8 × 5 = 40.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Is 11 a factor of 176? Give a reason.
176 ÷ 11 = 16, a whole number.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
What is the greatest common factor of 20 and 42?
The GCF is the largest whole number dividing both with no remainder: 2.
Scoring (2 points): Award one point for a sound method and one for the correct result.
List all positive factors of 44.
Each factor divides 44 with no remainder; pair divisors with their matching quotients.
Scoring (2 points): Award one point for a sound method and one for the correct result.
What is the greatest common factor of 14 and 15?
The GCF is the largest whole number dividing both with no remainder: 1.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the least positive common multiple of 6 and 5.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 30.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Classify 45 as prime, composite, or neither.
The positive factors are 1, 3, 5, 9, 15, 45. Prime means exactly two positive factors; 1 is neither prime nor composite.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the 5th positive multiple of 5.
5 × 5 = 25.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Classify 60 as prime, composite, or neither.
The positive factors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. Prime means exactly two positive factors; 1 is neither prime nor composite.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the 5th positive multiple of 7.
7 × 5 = 35.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Is 3 a factor of 33? Give a reason.
33 ÷ 3 = 11, a whole number.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
What is the greatest common factor of 18 and 15?
The GCF is the largest whole number dividing both with no remainder: 3.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the least positive common multiple of 7 and 14.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 14.
Scoring (2 points): Award one point for a sound method and one for the correct result.
What is the greatest common factor of 12 and 24?
The GCF is the largest whole number dividing both with no remainder: 12.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the least positive common multiple of 10 and 9.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 90.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Classify 31 as prime, composite, or neither.
The positive factors are 1, 31. Prime means exactly two positive factors; 1 is neither prime nor composite.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the 3rd positive multiple of 13.
13 × 3 = 39.
Scoring (2 points): Award one point for a sound method and one for the correct result.
List all positive factors of 80.
Each factor divides 80 with no remainder; pair divisors with their matching quotients.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find the 4th positive multiple of 11.
11 × 4 = 44.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Is 12 a factor of 60? Give a reason.
60 ÷ 12 = 5, a whole number.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Is zero a multiple of 12? Explain using multiplication.
12 × 0 = 0. A multiple includes zero when the multiplier is a whole number.
Scoring (4 points): Award one point for a sound method and one for the correct result.
Find the least positive common multiple of 14 and 5.
List positive multiples of each or use prime factors to locate the first shared positive multiple: 70.
Scoring (2 points): Award one point for a sound method and one for the correct result.
List all positive factors of 108.
Each factor divides 108 with no remainder; pair divisors with their matching quotients.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Sam claims 24 is prime because it is not divisible by 5. Explain why that test is insufficient, and classify 24.
Not being divisible by 5 proves nothing by itself. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. A prime has exactly two positive factors.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
What is the greatest common factor of 10 and 30?
The GCF is the largest whole number dividing both with no remainder: 10.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Is zero a multiple of 11? Explain using multiplication.
11 × 0 = 0. A multiple includes zero when the multiplier is a whole number.
Scoring (4 points): Award one point for a sound method and one for the correct result.