This week’s Micro Softy, The Bookkeeper’s Trick, relates in a way to the solution to last week’s puzzle because it involves a number that is divisible by 9.  So I’m going to break with tradition and talk about the solution to last week’s puzzle before presenting this week’s puzzle.

An interesting side note: The word bookkeeper has three groups of repeated letters in a row. I can’t think of any other word that has more. Can you?

Okay. Back to the solution to last week’s puzzle.

Solution to Micro Softy 42: Bacteria Multiply by Dividing

Show, without dividing, multiplying or using a calculator, that this number

x=810,180,810,333,180,810,180

is divisible by 2,3,4,5,6,9,10,12,15,18 20, and 30. Here are the answers.

• 2: x is an even number ending in a 0 so is divisible by 2.
• 3: Add up all the digits in x.

8+1+0+1+8+0+8+1+0+3+3+3+1+8+0+8+1+0+1+8+0 = 63

      and

6+3 = 9.

Image Credit: Fauzi - Adobe Stock

If adding up digits like this results in a 3, 6 or 9, the number is divisible by 3.

• 4: A number is divisible by 4 if its last two digits, added together, are divisible by 4. The last two digits of x are 80 which is divisible by 4. (Interesting sidenote: Every US presidential election occurs in a year divisible by 4.)
• 5: If the last digit of a number is either a 0 or a 5, the number is divisible by 5. The last digit of x is 0 so x is divisible by 5.
• 6: Since x is divisible by both 2 and 3, it is divisible by 2×3=6.
• 9: When looking at divisibility by 3 above, we repeatedly added the digits of x and got 9. If adding digits like this ends up being 9, the number is divisible by 9. Thus x is divisible by 9.
• 10: If x ends in a zero, it is divisible by 10.
• 12: Since x is divisible by 3 and 4, it is divisible by 12=3×4.
• 15: Since x is divisible by 3 and 5, it is divisible by 15=3×5.
• 18: Since x is divisible by 9 and 2, it is divisible by 18=9×2.
• 20: Since x is divisible by 4 and 5, it is divisible by 20=4×5.
• 30: Since x is divisible by 3 and 10, it is divisible by 30=3×10.

Note: We can’t conclude x is divisible by 27 because it is divisible by 3 and 9 because 3 and 9 are not relatively prime.

Incidentally, the number y=9,007,199,254,740,881 is not divisible by any of the numbers on the list above because y, while very large, is a prime number.

This Week’s Micro Softy 43:  The Bookkeeper’s Trick 

This one is about things that don’t add up.

Before describing the puzzle, I want to air a pet peeve. It’s because I’m a snooty engineer who uses a lot of high-level math.

When people get a wrong answer in adding or percentages, they often say “I’m bad at math.”

Bullpucky! You’re bad at arithmetic, which is just a teeny piece of the big, beautiful world of math.

So don’t say “I’m bad at math” if you don’t even know the difference between a quadratic equation and a differential equation. By elevating your ignorance at a higher level, you are bragging. Stay humble at your level. Say, rather, “I’m bad at arithmetic.”

OK. I feel better having got this off my chest.

Now here’s the Micro Softy:

It’s 1932. Bookkeepers wearing green visors hunch over small desks and spend their days adding up long lists of numbers in their heads and entering them on paper by hand. Bookkeepers Tim and Jim added up the same list of numbers but got different answers. Tim got 2,359,225 and Jim got 2,359,315.

Quinn, the supervisor, looked at the sums.

“The difference between the results is 90. And 90 is divisible by 9” he offered. “Chances are that you transposed two adjacent digits somewhere in your list of numbers.”

Jim and Tim went to Jim’s desk and took out a sheet of paper.

“Let’s see if this works if there is just one number on the list,” said Jim. He chose the number 25,795 and transposed the second and third digits to 27,595. The difference between the two numbers was 1800.

“Wow!” says Tim. “And 1800 is divisible by 9!”

Jim and Tim looked at the list of numbers they had added and discovered that Quinn was right. One of the numbers on their two lists had adjacent digits transposed.

Can you explain why this bookkeeper’s trick works?

The Micro Softy is a weekly feature of Mind Matters News. Here are the links to all the puzzles and answers to date:

Monday Micro Softy 42: Bacteria multiply by dividing. But our puzzlers must solve this math question without doing either. And no calculators allowed. A hint for solving Micro Softy 41: Dividing an unusually shaped piece of land among four sons rather than three is easier if you think in terms of fractals.

Image Credit: PRASERT - Adobe Stock

Monday Micro Softy 41: King David’s sons’ puzzling inheritance. Solomon must figure out how to divide an oddly shaped piece of land equally among his brothers and himself. In Micro Softy 40, the critical question is not whether the watchman’s prophecy comes true or not; it’s how he came to learn the information anyway.

Monday Micro Softy 40: The fate of a false prophet. He wasn’t actually fired for being a false prophet about the bomb threat but for something that his prophecy unintentionally revealed. The solution to Micro Softy 39 lies in considering an alternative possible meaning of a word commonly used in sports. You will also find links to Micro Softies 35 through 39 and their answers here as well.

And here are links to all the earlier Micro Softies

Monday Micro Softy 29: A funeral lament in four lines. The funeral director was puzzled by Dan’s description of his relationship to the deceased but there was no question that his grief was sincere Here, you will also find links to Microsofties 22 through 29.

Monday Micro Softy 21: Finding More of the Deadly Fentanyl Pills. There, you will also find links to Microsofties 11 through 20 as well.

At Monday Micro Softy 11: What Happened to That Other Dollar?, you will find links to the first ten Micro Softies. Have fun!