Level 5 Puzzle

Divisibility Tricks Applied to 693

/ ivasallay / Leave a comment
  • 693 is a composite number.
  • Prime factorization: 693 = 3 x 3 x 7 x 11, which can be written 693 = (3^2) x 7 x 11
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 693 has exactly 12 factors.
  • Factors of 693: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, 693
  • Factor pairs: 693 = 1 x 693, 3 x 231, 7 x 99, 9 x 77, 11 x 63, or 21 x 33
  • Taking the factor pair with the largest square number factor, we get √693 = (√9)(√77) = 3√77 ≈ 26.324893

Some quick divisibility tricks applied to the number 693:

  1. Every counting number is divisible by 1
  2. 693 is not even so it isn’t divisible by 2
  3. Every digit of 693 is divisible by 3, so 693 is divisible by 3
  4. Since it isn’t divisible by 2, it isn’t divisible by 4
  5. 693 doesn’t end in a 5 or 0, so it’s not divisible by 5
  6. 693 is divisible by 3 but not by 2 so it isn’t divisible by 6
  7. 69 – 2(3) = 63, a multiple of 7 so 693 is divisible by 7
  8. Since it isn’t divisible by 2 or 4, it can’t be divisible by 8
  9. 6 + 9 + 3 = 18, a multiple of 9 so 693 is divisible by 9
  10. Since the last digit of 693 isn’t 0, it is not divisible by 10
  11. 6 – 9 + 3 = 0, so 693 is divisible by 11

The divisibility tricks that worked on the number 693 are quite easy to see on the outside of this factor cake.

Here is today’s factoring puzzle:

693 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

———————————————————————————

693 is a palindrome in several bases:

  • 1010110101 BASE 2; note 1(512) + 0(256) + 1(128) + 0(64) + 1(32) + 1(16) + 0(8) + 1(4) + 0(2) + 1(1) = 693
  • 3113 BASE 6; note 3(216) + 1(36) + 1(6) + 3(1) = 693
  • 414 BASE 13; note 4(169) + 1(13) + 4(1) = 693
  • 313 BASE 15; note 3(225) + 1(15) + 3(1) = 693

———————————————————————————

693 Logic

Ricardo tweeted his work for this puzzle, too.

//platform.twitter.com/widgets.js

686 and Level 5

/ ivasallay / Leave a comment

686 is divisible by 2, not by 3, and not by 5. Is it divisible by 7? Let’s try a divisibility rule for 7’s: 68 – 2(6) = 56, a multiple of 7. Yes, 686 is divisible by 7.

686 is a palindrome in two bases:

  • 686 in base 10; note that 6(100) + 8(10) + 6(1) = 681
  • 222 in base 18; note that 2(324) + 2(18) + 2(1) = 681

686 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

—————————————————————————————————

  • 686 is a composite number.
  • Prime factorization: 686 = 2 x 7 x 7 x 7, which can be written 686 = 2 x (7^3)
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 x 4 = 8. Therefore 686 has exactly 8 factors.
  • Factors of 686: 1, 2, 7, 14, 49, 98, 343, 686
  • Factor pairs: 686 = 1 x 686, 2 x 343, 7 x 98, or 14 x 49
  • Taking the factor pair with the largest square number factor, we get √686 = (√49)(√14) = 7√14 ≈ 26.1916017.

Here is a little cake to simplify √686:

686 cake

Just take the square root of everything on the outside of the cake and multiply them together to get √686 = (√2)(√7)(√49) = 7√14

—————————————————————————————————

686 Logic

678 and Level 5

/ ivasallay / Leave a comment

678 is made from 3 consecutive numbers so it is divisible by 3. The middle number of those 3 consecutive numbers is not divisible by 3, so 678 is NOT divisible by 9.

678 is the hypotenuse of  Pythagorean triple 90-672-678. What is the greatest common factor of those 3 numbers?

678 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

—————————————————————————————————

  • 678 is a composite number.
  • Prime factorization: 678 = 2 x 3 x 113
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 678 has exactly 8 factors.
  • Factors of 678: 1, 2, 3, 6, 113, 226, 339, 678
  • Factor pairs: 678 = 1 x 678, 2 x 339, 3 x 226, or 6 x 113
  • 678 has no square factors that allow its square root to be simplified. √678 ≈ 26.038433.

—————————————————————————————————

678 Logic

670 and Level 5

/ ivasallay / Leave a comment

Because 5 is one of its factors, 670 is the hypotenuse of the Pythagorean triple 402-536-670. Which factor of 670 is the greatest common factor of those three numbers?

670 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

—————————————————————————————————

  • 670 is a composite number.
  • Prime factorization: 670 = 2 x 5 x 67
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 670 has exactly 8 factors.
  • Factors of 670: 1, 2, 5, 10, 67, 134, 335, 670
  • Factor pairs: 670 = 1 x 670, 2 x 335, 5 x 134, or 10 x 67
  • 670 has no square factors that allow its square root to be simplified. √670 ≈ 25.884358.

—————————————————————————————————

670 Logic

Thank you, Ricardo, for tweeting the solution:

//platform.twitter.com/widgets.js

663 Peanut Butter Kiss Puzzle

/ ivasallay / Leave a comment

It’s easy to see that 663 is divisible by three.

That may not be very interesting at all, but…..

Because 13 and 17, 663’s other two prime factors, have a remainder of one when each is divided by four, 663 is the hypotenuse of FOUR Pythagorean triples. Can you find the greatest common factor of each triple?

  • 420-513-663
  • 63-660-663
  • 255-612-663
  • 312-585-663

—————————————————————————————————

Art is not one of my strongest skills, but here is my attempt to make a puzzle that looks like a peanut butter kiss, a traditional Halloween candy:

663 Puzzle Peanut Butter Kisses

Color can be inviting, but it can also be distracting. Here is the same puzzle without the added color:

663 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-26

—————————————————————————————————

  • 663 is a composite number.
  • Prime factorization: 663 = 3 x 13 x 17
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 663 has exactly 8 factors.
  • Factors of 663: 1, 3, 13, 17, 39, 51, 221, 663
  • Factor pairs: 663 = 1 x 663, 3 x 221, 13 x 51, or 17 x 39
  • 663 has no square factors that allow its square root to be simplified. √663 ≈ 25.748786.

—————————————————————————————————

663 Logic

Thank you for sharing the solution, Ricardo, and happy Halloween:

//platform.twitter.com/widgets.js

654 and Level 5

/ ivasallay / Leave a comment

654 is the sum of the fourteen prime numbers from 19 to 73.

654 is also the hypotenuse of the Pythagorean triple 360-546-654. Which of 654’s factors is the greatest common factor of those three numbers?

654 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

—————————————————————————————————

  • 654 is a composite number.
  • Prime factorization: 654 = 2 x 3 x 109
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 654 has exactly 8 factors.
  • Factors of 654: 1, 2, 3, 6, 109, 218, 327, 654
  • Factor pairs: 654 = 1 x 654, 2 x 327, 3 x 218, or 6 x 109
  • 654 has no square factors that allow its square root to be simplified. √654 ≈ 25.57342.

—————————————————————————————————

654 Logic

640 Fall Factor Trees and Level 5

/ ivasallay / 4 Comments

Because it is fall, and 640 has many factors, I decided to make factor trees using fall colors. Get out your rakes!

640 factor trees

There are many other possible factor trees for 640, but raking leaves can be a lot of work, so I only made two of them.

640 is the hypotenuse of the Pythagorean triple 384-512-640.

OEIS.org informs us that 640 = 16!!!!!!, but if you type 16!!!!!! into a calculator, you will get an error message as soon as you type !!.

16!!!!!! ≠ (((((16!)!)!)!)!)!

There are 6 !’s so 16!!!!!! = 16(16-6)(16-12) = 16 x 10 x 4.

Here is this week’s Level 5 puzzle:

640 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

—————————————————————————————————

  • 640 is a composite number.
  • Prime factorization: 640 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 640 = (2^7) x 5
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 640 has exactly 16 factors.
  • Factors of 640: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 128, 160, 320, 640
  • Factor pairs: 640 = 1 x 640, 2 x 320, 4 x 160, 5 x 128, 8 x 80, 10 x 64, 16 x 40, or 20 x 32
  • Taking the factor pair with the largest square number factor, we get √640 = (√64)(√10) = 8√10 ≈ 25.298221

—————————————————————————————————

640 Logic

633 and Level 5

/ ivasallay / Leave a comment
  • 633 = 210 + 211 + 212 (three consecutive numbers)
  • 633 = 209 + 211 + 213 (three consecutive odd numbers)
  • 633 = 199 + 211 + 223 (three consecutive prime numbers)

Each of the sums above has 3 numbers, and 3 is a prime factor of 633. The middle number in each of the sums is 211 which is the other prime factor of 633.

Each of its two prime factors is 104 away from their average, 107.

Thus (107^2) – (104^2) = 633

The numbers in 633’s other factor pair are 1 and 633, and they are each 316 away from their average, 317.

Thus (317^2) – (316^2) = 633

633 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

—————————————————————————————————

  • 633 is a composite number.
  • Prime factorization: 633 = 3 x 211
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 633 has exactly 4 factors.
  • Factors of 633: 1, 3, 211, 633
  • Factor pairs: 633 = 1 x 633 or 3 x 211
  • 633 has no square factors that allow its square root to be simplified. √633 ≈ 25.15949.

—————————————————————————————————

633 Logic

 

 

627 and Level 5

/ ivasallay / Leave a comment

Here are two divisibility rules applied to 627 that give a positive result:

  • 6 + 2 + 7 = 15, a multiple of 3, so 627 is divisible by 3.
  • 6 – 2 + 7 = 11, so 627 is divisible by 11.

627 can be expressed as the sum of consecutive counting numbers 4 different ways. The numbers in bold are in the middle of each sum:

  • 208 + 209 + 210 = 627
  • 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 = 627
  • 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 627
  • 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+ 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 = 627

The logic on this Level 5 puzzle gets a bit complicated right when it’s almost done:

627 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

—————————————————————————————————

  • 627 is a composite number.
  • Prime factorization: 627 = 3 x 11 x 19
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 627 has exactly 8 factors.
  • Factors of 627: 1, 3, 11, 19, 33, 57, 209, 627
  • Factor pairs: 627 = 1 x 627, 3 x 209, 11 x 57, or 19 x 33
  • 627 has no square factors that allow its square root to be simplified. √627 ≈ 25.039968.

—————————————————————————————————

627 Logic

620 and Level 5

/ ivasallay / Leave a comment

620 is the sum of the four prime numbers from 149 to 163, and it is also the sum of the eight prime numbers from 61 to 97.

620 is the hypotenuse of the Pythagorean triple 372-496-620. What is the greatest common factor of those three numbers?

Today’s puzzle may be a little more difficult than most Level 5 puzzles, but go ahead, embrace the challenge!

620 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

—————————————————————————————————

  • 620 is a composite number.
  • Prime factorization: 620 = 2 x 2 x 5 x 31, which can be written 620 = (2^2) x 5 x 31
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 620 has exactly 12 factors.
  • Factors of 620: 1, 2, 4, 5, 10, 20, 31, 62, 124, 155, 310, 620
  • Factor pairs: 620 = 1 x 620, 2 x 310, 4 x 155, 5 x 124, 10 x 62, or 20 x 31
  • Taking the factor pair with the largest square number factor, we get √620 = (√4)(√155) = 2√155 ≈ 24.899799

—————————————————————————————————

620 Logic