Question 1 How many multiples of 4 are between 10 and 250? Answer 1 Step 1 The multiples of 4 that lie between 10 and 250 are 12, 16, 20, ..., 248. Step 2 Using the formula for the 𝑛 n-th term of an arithmetic sequence: 𝑎 𝑛 = 𝑎 + ( 𝑛 − 1 ) 𝑑 a n =a+(n−1)d where: 𝑎 = 12 a=12 (the first term) 𝑑 = 4 d=4 (the common difference) 𝑎 𝑛 = 248 a n =248 (the last term) Substitute these values: 248 = 12 + ( 𝑛 − 1 ) × 4 248=12+(n−1)×4 248 − 12 = 4 𝑛 − 4 248−12=4n−4 236 = 4 𝑛 − 4 236=4n−4 4 𝑛 = 240 4n=240 𝑛 = 60 n=60 The number of multiples of 4 that lie between 10 and 250 is 60. Question 2 To find the least common multiple of a pair of numbers, 60 and 72, using the prime factors method. Answer 1 Given pair of numbers: 60, 72. Concept used: The least common multiple of two or more numbers is the smallest multiple that is completely divisible by the corresponding numbers, leaving no remainder. In the prime factors method, we find the prime factorization of each number, then use the highest powers of each prime. Prime factorization of 60: 2 2 × 3 × 5 2 2 ×3×5 Prime factorization of 72: 2 3 × 3 2 2 3 ×3 2 LCM: 2 3 × 3 2 × 5 = 360 2 3 ×3 2 ×5=360 Question 3 To find other numbers which are factors of a number whose factor is 18. Answer 1 Formula used: Find the factors of 18, and they are surely the factors of another number. Calculation: Any number which is a factor of 18 will also be a factor of any multiple of 18. For instance, the factors of 18 are 1, 2, 3, 6, 9, and 18. So, any number that is a multiple of 18 (e.g., 36, 54, etc.) will also have these as factors. Question 4 I am looking for 3 consecutive multiples of 7. The sum of the squares of the 2 smallest is 147 less than the square of the largest. The first multiple of 7 is 𝑥 − 7 x−7, the 2nd multiple of 7 is 𝑥 x, and the 3rd multiple of 7 is 𝑥 + 7 x+7. Find the numbers. Answer 1 Step 1 The three consecutive multiples of 7 are 𝑥 − 7 x−7, 𝑥 x, and 𝑥 + 7 x+7. According to the question, ( 𝑥 − 7 ) 2 + 𝑥 2 = ( 𝑥 + 7 ) 2 − 147 (x−7) 2 +x 2 =(x+7) 2 −147 Expanding and simplifying, 𝑥 2 − 14 𝑥 + 49 + 𝑥 2 = 𝑥 2 + 14 𝑥 + 49 − 147 x 2 −14x+49+x 2 =x 2 +14x+49−147 2 𝑥 2 − 14 𝑥 + 49 = 𝑥 2 + 14 𝑥 − 98 2x 2 −14x+49=x 2 +14x−98 𝑥 2 − 28 𝑥 + 147 = 0 x 2 −28x+147=0 Factoring, ( 𝑥 − 21 ) ( 𝑥 − 7 ) = 0 (x−21)(x−7)=0 So, 𝑥 = 21 x=21 or 𝑥 = 7 x=7. Step 2 When 𝑥 = 21 x=21: The consecutive multiples of 7 are 14 , 21 , 28 14,21,28. When 𝑥 = 7 x=7: The consecutive multiples of 7 are 0 , 7 , 14 0,7,14. Question 5 To find: the least common denominator of the fractions. 2 5 𝑥 2 + 3 𝑥 4 5x 2 2 + 4 3x 𝑥 2 𝑦 + 4 3 𝑥 2y x + 3x 4 Answer 1 The least common denominator (LCD) is defined as the least common multiple of the denominators of the set of fractions. Example 1: 2 5 𝑥 2 + 3 𝑥 4 5x 2 2 + 4 3x Prime factorization of each denominator: 5 𝑥 2 = 5 × 𝑥 × 𝑥 5x 2 =5×x×x 4 = 2 × 2 4=2×2 LCD: 5 × 𝑥 2 × 2 × 2 = 20 𝑥 2 5×x 2 ×2×2=20x 2 Example 2: 𝑥 2 𝑦 + 4 3 𝑥 2y x + 3x 4 Prime factorization of each denominator: 2 𝑦 = 2 × 𝑦 2y=2×y 3 𝑥 = 3 × 𝑥 3x=3×x LCD: 2 × 𝑦 × 3 × 𝑥 = 6 𝑥 𝑦 2×y×3×x=6xy Final Statement: The least common denominator is defined as the least common multiple of the denominators of the set of fractions.