Step 1: Prime factorise 216. Step 2: Find the GCF of 8 and 6. Step 3: Find the LCM of 12 and 18. Step 4: Find how many triangular numbers will form a sum of 36.

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Asked on May 17, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-17T11:22:43.571Z

Step 1: Prime factorise 216. We find the prime factors of 216 by dividing it by the smallest prime numbers. 216 ÷ 2 = 108 108 ÷ 2 = 54 54 ÷ 2 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, the prime factors of 216 are 2 × 2 × 2 × 3 × 3 × 3. In exponential form, this is 2^3 × 3^3. In set notation, the prime factors are the unique prime numbers involved. The prime factors of 216 are \2, 3\. Step 2: Find the GCF of 8 and 6. First, list the factors for each number. Factors of 8: 1, 2, 4, 8 Factors of 6: 1, 2, 3, 6 The common factors are 1 and 2. The greatest common factor (GCF) is the largest of these common factors. The GCF of 8 and 6 is 2. Step 3: Find the LCM of 12 and 18. We can find the LCM by listing multiples or using prime factorization. Using prime factorization: Prime factors of 12: 2^2 × 3^1 Prime factors of 18: 2^1 × 3^2 To find the LCM, we take the highest power of each prime factor present in either number. LCM = 2^2 × 3^2 = 4 × 9 = 36. The LCM of 12 and 18 is 36. Step 4: Find how many triangular numbers will form a sum of 36. Triangular numbers are generated by the formula T_n = (n(n+1))/(2). Let's list the first few triangular numbers: T_1 = (1(1+1))/(2) = 1 T_2 = (2(2+1))/(2) = 3 T_3 = (3(3+1))/(2) = 6 T_4 = (4(4+1))/(2) = 10 T_5 = (5(5+1))/(2) = 15 T_6 = (6(6+1))/(2) = 21 T_7 = (7(7+1))/(2) = 28 T_8 = (8(8+1))/(2) = 36 The 8th triangular number is 36. Therefore, one triangular number (the 8th one) forms a sum of 36. The number of triangular numbers is 1. Step 5: Find the next numbers in the sequences. i) 5, 10, 15, 20, 25, 30, ___, 40, 45, 50 This is an arithmetic sequence where each term increases by 5. The next number after 30 is 30 + 5 = 35. The sequence is 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. ii) 1, 3, ___, 7, 9, ___ This is an arithmetic sequence of odd numbers, where each term increases by 2. The number after 3 is 3 + 2 = 5. The number after 9 is 9 + 2 = 11. The sequence is 1, 3, 5, 7, 9, 11. iii) 1, 4, 9, 16, 25, ___ This sequence consists of perfect squares: 1^2, 2^2, 3^2, 4^2, 5^2. The next term in the sequence will be 6^2. 6^2 = 36. The sequence is 1, 4, 9, 16, 25, 36. That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-05-17T11:22:43.571Z

Step 1: Prime factorise 216. We find the prime factors of 216 by dividing it by the smallest prime numbers. 216 ÷ 2 = 108 108 ÷ 2 = 54 54 ÷ 2 = 27 27 ÷ 3 = 9 9 ÷ 3 = 3 3 ÷ 3 = 1 So, the prime factors of 216 are 2 × 2 × 2 × 3 × 3 × 3. In exponential form, this is 2^3 × 3^3. In set notation, the prime factors are the unique prime numbers involved. The prime factors of 216 are \2, 3\. Step 2: Find the GCF of 8 and 6. First, list the factors for each number. Factors of 8: 1, 2, 4, 8 Factors of 6: 1, 2, 3, 6 The common factors are 1 and 2. The greatest common factor (GCF) is the largest of these common factors. The GCF of 8 and 6 is 2. Step 3: Find the LCM of 12 and 18. We can find the LCM by listing multiples or using prime factorization. Using prime factorization: Prime factors of 12: 2^2 × 3^1 Prime factors of 18: 2^1 × 3^2 To find the LCM, we take the highest power of each prime factor present in either number. LCM = 2^2 × 3^2 = 4 × 9 = 36. The LCM of 12 and 18 is 36. Step 4: Find how many triangular numbers will form a sum of 36. Triangular numbers are generated by the formula T_n = (n(n+1))/(2). Let's list the first few triangular numbers: T_1 = (1(1+1))/(2) = 1 T_2 = (2(2+1))/(2) = 3 T_3 = (3(3+1))/(2) = 6 T_4 = (4(4+1))/(2) = 10 T_5 = (5(5+1))/(2) = 15 T_6 = (6(6+1))/(2) = 21 T_7 = (7(7+1))/(2) = 28 T_8 = (8(8+1))/(2) = 36 The 8th triangular number is 36. Therefore, one triangular number (the 8th one) forms a sum of 36. The number of triangular numbers is 1. Step 5: Find the next numbers in the sequences. i) 5, 10, 15, 20, 25, 30, ___, 40, 45, 50 This is an arithmetic sequence where each term increases by 5. The next number after 30 is 30 + 5 = 35. The sequence is 5, 10, 15, 20, 25, 30, 35, 40, 45, 50. ii) 1, 3, ___, 7, 9, ___ This is an arithmetic sequence of odd numbers, where each term increases by 2. The number after 3 is 3 + 2 = 5. The number after 9 is 9 + 2 = 11. The sequence is 1, 3, 5, 7, 9, 11. iii) 1, 4, 9, 16, 25, ___ This sequence consists of perfect squares: 1^2, 2^2, 3^2, 4^2, 5^2. The next term in the sequence will be 6^2. 6^2 = 36. The sequence is 1, 4, 9, 16, 25, 36. That's 2 down. 3 left today — send the next one.

Step 1: Prime factorise 216. Step 2: Find the GCF of 8 and 6. Step 3: Find the LCM of 12 and 18. Step 4: Find how many triangular numbers will form a sum of 36.

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Step 1: Prime factorise 216. Step 2: Find the GCF of 8 and 6. Step 3: Find the LCM of 12 and 18. Step 4: Find how many triangular numbers will form a sum of 36.

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