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
a. 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.
b. Create your own “number trick.”
2. Calculation “Trick” 1
a. 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:
45 2 = __2025__
652 = __4225___
752 = __5625___
Prove it! Let the number be 10t + u, so it appears as tu.
b. 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!
3. Calculation “Trick” 2
a. Perform the following without a calculator in a total of 90 seconds or less.
i. (31)(29) = (30 + 1)(30 – 1) = 900 – 1 = 899
ii. (1002)(998) = (1000 + 2)(1000 – 2) = 1,000,000 – 4 = 999, 996
iii. (97)(103) = (100 – 3)(100 + 3) = 10,000 – 9 = 9991
iv. (12,100)(11,900) = (12,000 + 100)(12,000 – 100) = 144,000,000 – 10,000
= 143,990,000
v. (3996)(4004) = (4000 – 4)(4000 + 4) = 16,000,000 – 16 = 15,999,984
b. Explain!! If the two numbers are equally distant from a “nice” value, use the
factoring form for a difference of squares!
4. Calculation “Trick” 3
a. Watch the first part of the following: http://www.youtube.com/watch?v=I9tgYnPNaw. 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.
b. To cube a two-digit number, do the following:
To find (tu )3 , the ones digit will be u 3 , the tens digit will be 3tu 2 , the hundreds
digit will be 3t 2u and the thousands digit will be t 3 . If any of those are two digit
numbers (or more), carry the tens digit over to the next place. For example:
123  13 3 12  2 3 1  2 2 23
= 1
6
12
8
= 1728
Try to find 353 = 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