The Largest Product

Transcript

The Largest ProductThe goal is to take three given digits and arrange them into a multiplication problem that looks like this: [ ][ ] × [ ] to get the largest possible answer. For digits 2, 3, 5 We list all possible cases The Logic: How to Compare Without Multiplying Everything We could just calculate all six, but mathematicians love finding shortcuts! The book groups them into pairs based on their single-digit multiplier: Multiplier is 2: 35×2 and 53×2 Multiplier is 3: 25×3 and 52×3 Multiplier is 5: 23×5 and 32×5 In each pair, the one with the bigger 2 -digit number will obviously win. For example - 53×2 is clearly larger than 35×2. So, the real competition is only between the champions of each group: 53 × 2 52 × 3 32 × 5 Now, Right away, 52×3 (which is over 150) beats 53×2 (which is about 100). So the final showdown is between 52×3 and 32×5. To see which is bigger, we expand the numbers using place value: 32 × 5 becomes = (3 × 10 + 2) × 5 = 3 × 10 × 5 + 2 × 5 52 × 3 becomes = (5 × 10 + 2) × 3 = 5 × 10 × 3 + 2 × 3 Look closely at the first part of both equations: (3 × 10 × 5) and (5 × 10 × 3) They both equal 150! Because those main chunks are identical, the winner is decided entirely by the second chunk (the ones place). For the top one: 2 × 5=10 For the bottom one: 2 × 3=6 Since 10 is greater than 6, 32 × 5 is the absolute largest product you can make with the digits 2, 3, and 5 Algebraic Proof This gives us our Golden Rule for the Largest Product: To get the biggest answer, the largest digit must be the multiplier, and the other two digits must be arranged in decreasing order (biggest to smallest) to form the 2-digit number.