FUN WITH NUMBERS – key

ARML Practice Problems – October 24, 2010

This is going to be a mini-PQ. You will play with some ideas and then try to go toward proving that your inductive reasoning was correct.

  1. Supereasy
  2. Do the following trick – play the game!

Choose a number

Add 5

Double the result

Subtract 4

Divide by 2

Subtract the original number.

You got an answer of 3, didn’t you?

Prove it! Use an x as the number and run through the algebra. The final value is not dependent upon the beginning value.

  1. Create your own “number trick.”
  1. Calculation “Trick” 1
  2. To easily square a two-digit number, ending in 5, do the following:

Let the number be a5. Multiply a by (a + 1) and then append 25 on the end. So, 35 = (3 x 4)25 = 1225

Try it on the following:

= __2025__

= __4225___

= __5625___

Prove it! Let the number be 10t + u, so it appears as tu.

  1. Does this work on three-digit numbers, ending in 5? Four-digit? N-digit?

The proof above indicates that this will happen, no matter how many digits!

  1. Calculation “Trick” 2
  2. Perform the following without a calculator in a total of 90 seconds or less.
  3. (31)(29) = (30 + 1)(30 – 1) = 900 – 1 = 899
  4. (1002)(998) = (1000 + 2)(1000 – 2) = 1,000,000 – 4 = 999, 996
  5. (97)(103) = (100 – 3)(100 + 3) = 10,000 – 9 = 9991
  6. (12,100)(11,900) = (12,000 + 100)(12,000 – 100) = 144,000,000 – 10,000 = 143,990,000
  7. (3996)(4004) = (4000 – 4)(4000 + 4) = 16,000,000 – 16 = 15,999,984
  1. Explain!! If the two numbers are equally distant from a “nice” value, use the factoring form for a difference of squares!
  1. Calculation “Trick” 3
  2. Watch the first part of the following: Stop and explain how the “trick” works.

Let the number A = 100 + a. Then A2 = (100 + a)2 = 10,000 + 100a + 100a + a2. So, A2 will be 100A + 100a + a2, so the trick works.

  1. To cube a two-digit number, do the following:

To find , the ones digit will be , the tens digit will be , the hundreds digit will be and the thousands digit will be . If any of those are two digit numbers (or more), carry the tens digit over to the next place. For example:

Try to find = 33 3x32x5 3x3x52 53 = 27 135 225 125 = 42,875

Explain why this works! Will it work for the cubing of a three-digit number? Four-digit? N-digit? Because (10t + u)3 = 1000t3 + 300t2u + 30tu2 + u3