by Bill Graham

Calculator discourse

Sad to say, some students don’t seem capable of mastering these basic skills. When they get to seventh or eighth grade, they are still incapable or unmotivated to memorize 6 × 9 = 54. Then, I hand them a calculator and let them learn other areas of mathematics in spite of their shortcoming. That’s my opinion. What’s yours?

First calculator puzzle

Here is your first calculator puzzle. Enter the number 12,345,679 on your calculator. Now multiply it by 45. What is the result? All fives I hope. Suppose your favorite number is eight. Multiply the original 12,345, 679 by 72 and you should have all eights. What’s the rule to determine what to multiply by if you want all threes or all sevens? Do you see why this works?

Remainder discourse

Most students view doing long division with remainders as one of their least favorite things. It’s right up (down) there with fractions. If you were given a large random number and asked to divide it by five, would you expect a remainder or not?

You would not get a remainder if the original number is a multiple of five (a number than ends in five or zero) but would get a remainder for all other numbers. So, you would expect a remainder 80 percent of the time. If you divided by seven, you would expect a whole number for an answer every multiple of seven, or in other words, you would expect a remainder about six-sevenths of the time.

Another puzzle

Just lucky I guess. OK, let’s keep going.

Now divide the answer on your calculator by 11. Surely we’ll get a remainder this time. Nope, exactly 4,511. Finally, divide that answer by 13 and check for a remainder. Whoa! Not only is there no remainder again, but the answer is the three digit number you started with, namely 347.

If you try this same procedure starting with any other repeated three-digit number, the answers will always be whole numbers, and the final quotient will always be the number you started with. How come?