This week’s puzzles for you to solve
70 is a composite number. 70 = 1 x 70, 2 x 35, 5 x 14, and 7 x 10. Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70. Prime factorization: 70 = 2 x 5 x 7.
Sometimes 70 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we only use 70 = 7 x 10 in the puzzles.
Here are the factors for today’s puzzle:
The number 6 appears twice as a clue in today’s puzzle. It shouldn’t be difficult to figure out why I put “Six!” in this post’s title.
This week’s puzzles for you to solve
69 is a composite number. 69 = 1 x 69 or 3 x 23. Factors of 69: 1, 3, 23, 69. Prime factorization: 69 = 3 x 23.
69 is never a clue in the FIND THE FACTORS puzzles.
Here are the factors for today’s puzzle:
68 is a composite number. 68 = 1 x 68, 2 x 34, or 4 x 17. Factors of 68: 1, 2, 4, 17, 34, 68. Prime factorization: 68 = 2 x 2 x 17, which can also be written 2² x 17.
68 is never a clue in the FIND THE FACTORS puzzles.
All Previously Published Puzzles
I survived my daughter’s wedding. The last of our out-of-state visitors will be going home tomorrow. AND my computer that broke and spent a week in the shop is working again. I will publish six new puzzles next Monday.
Shortly after I started blogging, Steve Morris found me and became a leader in my cheering section. Even after the number of HIS followers grew exponentially, he still finds some time to show me support.
Here he writes an entertaining and encouraging post titled How To Get More Blog Followers that I think all bloggers should read. It has a mathematical element to it and could help you increase the number of people who read your blog even if your own mother isn’t one of them.
Now here’s a little about the number 67:
How do we know that 67 is a prime number? If 67 were not a prime number, then it would be divisible by at least one prime number less than or equal to √67 ≈ 8.2. Since 67 cannot be divided evenly by 2, 3, 5, or 7, we know that 67 is a prime number.
67 is never a clue in the FIND THE FACTORS puzzles.
66 is a composite number. 66 = 1 x 66, 2 x 33, 3 x 22, or 6 x 11. Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66. Prime factorization: 66 = 2 x 3 x 11.
Sometimes 66 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, the puzzle only uses 6 x 11 = 66.
I like this fractal animation very much! It really goes on forever!
65 = (1 + 2 + 3 +. . . + 23 + 24 + 25)/5 so it is the magic sum of a 5 × 5 magic square. I solved this 5×5 magic square game using a time-honored algorithm. Notice that 11, 12, 13, 14, and 15 are on a diagonal. Actually, all of the numbers are sort of on a diagonal if you follow the instructions given here. Also notice that the first number, 1, is in the center of the top row and the last number, 25, is in the center of the bottom row.
You can try to solve the square, too. Just click on the magic square image above to go to a website where you can try these or other combinations of numbers. You will know you have solved the magic square when the sum of each row, column, and diagonal totals 65.
65 is also the sum of two squares two different ways:
That makes 65 the hypotenuse of FOUR Pythagorean triples:
65 is a composite number. 65 = 1 x 65 or 5 x 13. Factors of 65: 1, 5, 13, 65. Prime factorization: 65 = 5 x 13.
65 – 56 = 9, a square number and 65+56 = 121, another square number. OEIS.org informs us that 65 and 56 are the smallest pair of mirror image numbers that produce a square number when either added or subtracted.
64 is a composite number, and it is 8 squared. 64 = 1 x 64, 2 x 32, 4 x 16, or 8 x 8. Factors of 64: 1, 2, 4, 8, 16, 32, 64. Prime factorization: 64 = 2 x 2 x 2 x 2 x 2 x 2, which can also be written 64 = 2⁶.
Since √64 = 8, a whole number, 64 is a perfect square.
Sometimes 64 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, when it is a clue, the factors are always 8 x 8.
64 is in this cool pattern:
All Previously Published Puzzles
This solution is complicated, but here is a logical explanation:
There are only two 3s available, so at most one 9 can be 3 x 3.
Can both 9s be 9 x 1? If both were, then both 1s would be used, leaving 8 = 2 x 4 and 10 = 2 x 5 as the only possibilities for 8 and 10. But if both 9s were 9 x 1, then the 18 in the same row as the 9 would have to be 18 = 9 x 2, and that would use too many twos. Therefore, EXACTLY ONE of the nines can use 9 = 9 x 1.
Since one 9 = 3 x 3, using both 3s, 30 must equal 5 x 6. And since both 30 and 50 must use both 5s, 10 must equal 10 x 1.
Since 10 = 10 x 1 and a 9 = 9 x 1, 8 must equal 2 x 4. So now finally we can begin filling out the puzzle:
63 is a composite number. 63 = 1 x 63, 3 x 21, or 7 x 9. Factors of 63: 1, 3, 7, 9, 21, 63. Prime factorization: 63 = 3 x 3 x 7, which can also be written 63 = 3² x 7.
Sometimes 63 is a clue in the FIND THE FACTORS puzzles. Even though it has other factors, the only multiplication fact we use is 7 x 9 = 63.
All Previously Published Puzzles
62 is a composite number. 62 = 1 x 62 or 2 x 31. Factors of 62: 1, 2, 31, 62. Prime factorization: 62 = 2 x 31.
62 is never a clue in the FIND THE FACTORS puzzles.
All Previously Published Puzzles
Here is the logic to solve the puzzle & a chart showing one way to get the solution:
9 has to be the factor of both the 27 and the 63. Therefore 3 x 6 = 18, (not 2 x 9).
3 is one of the factors of both the 18 and the 27. Therefore 9 x 1 = 9, (not 3 x 3).
How do we know that 61 is a prime number? If 61 were not a prime number, then it would be divisible by at least one prime number less than or equal to √61 ≈ 7.8. Since 61 cannot be divided evenly by 2, 3, 5, or 7, we know that 61 is a prime number.
61 is never a clue in the FIND THE FACTORS puzzles.
Wedding Bells are Going to Chime
My daughter is getting married in early March, and family will be coming in from out of town. I also need to study for a test at work. There is so much that needs to get done that hasn’t been getting done. I’m taking a month-long break from daily blogging and a few other things so I can catch up on the things that just can’t wait.
Over the last several weeks there has been several new people who have found this blog and have done some of the puzzles. Some of them have clicked on the older excel files to work on puzzles. All that clicking was probably inconvenient, but several people have still clicked anyway. Thank you for your interest and persistence. I want to make it easier to catch up on all those old puzzles.
To make it easier to access the older puzzles, I’ve put all previously published FIND THE FACTORS 1-10 and 1-12 puzzles in one excel file, All Previously Published Puzzles. The formatting isn’t printer friendly, but if you enable editing, you can type the factors directly onto the excel file. The answers are also in the file for easy comparison with your work. (Hopefully, I didn’t mess up any of the clues while I hurriedly created this large file. Please, tell me if you find any errors.)
While I catch up on all the things I need to do, I hope you will catch up a few things as well.