Playing with Numbers

This chapter treats numbers as objects to explore. You will learn to write numbers in general form: a two-digit number with tens digit a and units digit b is 10a + b, and a three-digit number with digits a, b and c is 100a + 10b + c. Using the general form, you will explain number games, for example why the sum of a two-digit number and its reverse is always divisible by 11, why their difference is divisible by 9, and why the difference between a three-digit number and its reverse is divisible by 99. You will also solve letters-for-digits puzzles, called cryptarithms, such as 3A + 25 = B2, using the rules that each letter stands for exactly one digit and that the first digit of a number cannot be zero. Finally, you will see why the familiar tests of divisibility by 10, 5, 2, 9 and 3 work, and use them to find missing digits in numbers.

Detailed Worksheet: Playing with Numbers Section A - Definitions (10 marks) 1. What is the general form of a two-digit number? Write 57 and 90 in general form. (2 marks) 2. Write the general form of a three-digit number with digits a, b and c. Write 351 and 608 in this form. (2 marks) 3. State the tests of divisibility by 2, 5 and 10. (2 marks) 4. State the tests of divisibility by 3 and 9. Is every number divisible by 3 also divisible by 9? Give an example. (2 marks) 5. What is a cryptarithm? State the two rules followed while solving letters-for-digits puzzles. (2 marks) Section B - Calculations and Applications (15 marks) 6. Show that 47 + 74 is divisible by 11. Using the general form 10a + b, prove that the sum of any two-digit number and the number obtained by reversing its digits is divisible by 11, and state the quotient. (3 marks) 7. Find 83 - 38 and divide it by 9. Prove, using the general form, that the difference between a two-digit number and its reverse is always divisible by 9, and state the quotient when the tens digit is greater. (3 marks) 8. Find 632 - 236 and show that it is divisible by 99. Prove the general result for a three-digit number 100a + 10b + c with a greater than c. (3 marks) 9. (i) If 21y5 is a multiple of 9, where y is a digit, find y. (ii) If 31z5 is a multiple of 3, where z is a digit, find all possible values of z. (3 marks) 10. Solve the cryptarithms, explaining your reasoning: (i) 3A + 25 = B2 (ii) A1 + 1B = B0. (3 marks) Section C - Diagrams (10 marks) 11. Draw a flowchart for testing whether a number is divisible by 9. Use the place-value split 100a + 10b + c = 99a + 9b + (a + b + c) to explain why the test works, and apply your flowchart to 7317. (4 marks) 12. Draw a flowchart for the game "choose a two-digit number, reverse its digits, add the two numbers and divide by 11". Show the steps for 64 and state what the final answer always equals. (3 marks) 13. Draw a Venn diagram with two circles showing the numbers from 1 to 30 that are divisible by 2 and those divisible by 3. Which numbers lie in the overlapping region, and what are they multiples of? (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Take the three-digit number 417 and form the numbers 174 and 741 by moving the digits in a cycle. Add the three numbers and show that the sum is divisible by 37. Prove that this is true for every three-digit number by writing all three numbers in general form. (5 marks) 15. Solve the multiplication cryptarithms, explaining each step: (i) BA x B3 = 57A (ii) AB x 6 = BBB. (5 marks) 16. Using the general form 1000a + 100b + 10c + d of a four-digit number, explain why the number is divisible by 3 when the sum of its digits is divisible by 3. Hence explain why a number and the sum of its digits leave the same remainder when divided by 9. Check with 4567. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Use the general form of numbers in every proof and explain each step of the cryptarithms.