John has swimming lessons every fifth day and diving lessons every third day. If he had a swimming lesson and a diving lesson on 17 October 2019, when will be the next date on which he has both swimming and diving lessons?

Question 1 Step 1: John has swimming lessons every 5 days and diving lessons every 3 days. Both lessons occur together every LCM(5, 3) days. LCM(5, 3) = (5 × 3)/((5, 3)) = (15)/(1) = 15 He will have both lessons on the 15^th day. 15^th day Question 2 Step 1: The greatest number of parties, where each uses the same number of white and orange balloons, is the GCD of 24 and 16. Euclidean algorithm: 24 = 1 × 16 + 8 16 = 2 × 8 + 0 GCD(24, 16) = 8. Step 2: Number of white balloons per party: 24 ÷ 8 = 3. Number of orange balloons per party: 16 ÷ 8 = 2. The greatest number of parties is 8. 8 Question 3: Write numbers in standard form (scientific notation) correct to 3 significant figures. 3(i) 2.8745760 Step 1: Identify first 3 significant figures: 2, 8, 7. Fourth digit is 4 (<5, no rounding). 2.87 × 10^0 or simply 2.87 2.87 3(ii) 2.38575 × 10^6 Step 1: Mantissa 2.38575, first 3 significant figures: 2, 3, 8. Fourth digit 5 followed by 7 (>0), round up 8 to 9. 2.39 × 10^6 2.39 × 10^6 3(iii) 0.00002385 × 10^-6 Step 1: Simplify the number. 0.00002385 = 2.385 × 10^-5 2.385 × 10^-5 × 10^-6 = 2.385 × 10^-11 Step 2: Round mantissa 2.385 to 3 significant figures. First 3: 2, 3, 8. Fourth digit 5, round up 8 to 9 (common convention for homework). 2.39 × 10^-11 2.39 × 10^-11