Level 6 Puzzle

922 Boo!

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Have you ever cut holes in a sheet, put it over your head, and jumped out in front of people as you hollered, “Boo!”? Today’s puzzle is made to look like a ghost. It’s a level 6, but don’t let that spook you! Attack the puzzle using logic, and after you solve it, you can claim to be a ghost-buster!

Print the puzzles or type the solution on this excel file: 10-factors-914-922

I think my ghost is cute, maybe not as cute as any that might knock on your door on Halloween, but still quite cute.

When 922 floats around in a different base, you may think you’re seeing an apparition:

922 becomes 1234 in BASE 9 because 1(9³) + 2(9²) + 3(9¹) + 4(9º) = 922.
922 becomes palindrome 262 in BASE 20

922 is also the sum of the 18 prime numbers from 17 to 89.

922 = 29² + 9², so 922 is the hypotenuse of a Pythagorean triple:
522-760-922, which is the same as 2(29)(9), 29² – 9², 29² + 9².
That Pythagorean triple is also 2 times (261-380-461).

  • 922 is a composite number.
  • Prime factorization: 922 = 2 × 461
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 922 has exactly 4 factors.
  • Factors of 922: 1, 2, 461, 922
  • Factor pairs: 922 = 1 × 922 or 2 × 461
  • 922 has no square factors that allow its square root to be simplified. √922 ≈ 30.3644529

 

 

912 and Level 6

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912 is the sum of the ten prime numbers from 71 to 109.

It is also the sum of these four consecutive primes:

  • 223 + 227 + 229 + 233 = 912

912 is 192 in BASE 26 because 1(26²) + 9(26) + 2(1) = 912.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 912 is a composite number.
  • Prime factorization: 912 = 2 × 2 × 2 × 2 × 3 × 19, which can be written 912 = 2⁴ × 3 × 19
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 912 has exactly 20 factors.
  • Factors of 912: 1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912
  • Factor pairs: 912 = 1 × 912, 2 × 456, 3 × 304, 4 × 228, 6 × 152, 8 × 114, 12 × 76, 16 × 57, 19 × 48 or 24 × 38
  • Taking the factor pair with the largest square number factor, we get √912 = (√16)(√57) = 4√57 ≈ 30.1993377.

904 and Level 6

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30² + 2² = 904

That means 904 is the hypotenuse of a Pythagorean triple:

  • 120-896-904 which is 8 times (15-112-113)

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 904 is a composite number.
  • Prime factorization: 904 = 2 × 2 × 2 × 113, which can be written 904 = 2³ × 113
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 904 has exactly 8 factors.
  • Factors of 904: 1, 2, 4, 8, 113, 226, 452, 904
  • Factor pairs: 904 = 1 × 904, 2 × 452, 4 × 226, or 8 × 113
  • Taking the factor pair with the largest square number factor, we get √904 = (√4)(√226) = 2√226 ≈ 30.06659

895 and Level 6

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895 is the hypotenuse of Pythagorean triple 537-716-895, which is (3-4-5) times 179.

895 is also palindrome 292 in BASE 19 because 2(19²) + 9(19) + 2(1) = 895.

Print the puzzles or type the solution on this excel file: 12 factors 886-896

  • 895 is a composite number.
  • Prime factorization: 895 = 5 × 179
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 895 has exactly 4 factors.
  • Factors of 895: 1, 5, 179, 895
  • Factor pairs: 895 = 1 × 895 or 5 × 179
  • 895 has no square factors that allow its square root to be simplified. √895 ≈ 29.91655

885 and Level 6

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885 obviously is divisible by 5. Since it is exactly 3 numbers away from 888, it also can be evenly divided by 3 so adding up all its digits was not necessary to test for divisibility.

885 is the hypotenuse of a Pythagorean triple:

  • 531-708-885 which is (3-4-5) times 177

885 is palindrome 181 in BASE 26 because 1(26²) + 8(26) + 1(1) = 885

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 885 is a composite number.
  • Prime factorization: 885 = 3 × 5 × 59
  • 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 × 2 × 2 = 8. Therefore 885 has exactly 8 factors.
  • Factors of 885: 1, 3, 5, 15, 59, 177, 295, 885
  • Factor pairs: 885 = 1 × 885, 3 × 295, 5 × 177, or 15 × 59
  • 885 has no square factors that allow its square root to be simplified. √885 ≈ 29.74894956

874 and Level 6

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874 is the sum of the first 23 prime numbers:

  • 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 874

ALL of 874’s prime factors were included in that list.

Print the puzzles or type the solution on this excel file: 12 factors 864-874

  • 874 is a composite number.
  • Prime factorization: 874 = 2 × 19 × 23
  • 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 × 2 × 2 = 8. Therefore 874 has exactly 8 factors.
  • Factors of 874: 1, 2, 19, 23, 38, 46, 437, 874
  • Factor pairs: 874 = 1 × 874, 2 × 437, 19 × 46, or 23 × 38
  • 874 has no square factors that allow its square root to be simplified. √874 ≈ 29.56349.

860 and Level 6

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Print the puzzles or type the solution on this excel file: 10-factors-853-863

860 is the hypotenuse of a Pythagorean triple: 516-688-860, which is (3-4-5) times 172.

860 can be written as the sum of four consecutive prime numbers: 199 + 211 + 223 + 227 = 860

  • 860 is a composite number.
  • Prime factorization: 860 = 2 × 2 × 5 × 43, which can be written 860 = 2² × 5 × 43
  • 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 × 2 × 2 = 12. Therefore 860 has exactly 12 factors.
  • Factors of 860: 1, 2, 4, 5, 10, 20, 43, 86, 172, 215, 430, 860
  • Factor pairs: 860 = 1 × 860, 2 × 430, 4 × 215, 5 × 172, 10 × 86, or 20 × 43,
  • Taking the factor pair with the largest square number factor, we get √860 = (√4)(√215) = 2√215 ≈ 29.3257566

852 and Level 6

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Print the puzzles or type the solution on this excel file: 12 factors 843-852

I knew that 852 was divisible by 3 as soon as I typed it in a straight line on the number pad. Any 3 digit number that lies on a straight line on a number pad or a phone dial pad is divisible by 3. And in case you’ve ever wondered why the numbers on a number pad or calculator and the numbers on a phone dial pad are reversed, ABC News has the answer.

852 is 705 in BASE 11, and it is 507 in BASE 13.

852 is palindrome 1E1 in BASE 23 (E is 14 base 10) because 1(23²) +14(23¹) + 1(23º) = 852.

852 is the sum of consecutive prime numbers 421 and 431.

852 is also the 24th pentagonal number because (3⋅24² – 24)/2 = 852

  • 852 is a composite number.
  • Prime factorization: 852 = 2 × 2 × 3 × 71, which can be written 852 = 2² × 3 × 71
  • 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 × 2 × 2 = 12. Therefore 852 has exactly 12 factors.
  • Factors of 852: 1, 2, 3, 4, 6, 12, 71, 142, 213, 284, 426, 852
  • Factor pairs: 852 = 1 × 852, 2 × 426, 3 × 284, 4 × 213, 6 × 142, or 12 × 71,
  • Taking the factor pair with the largest square number factor, we get √852 = (√4)(√213) = 2√213 ≈ 29.189039

847 Sending Love to My Sister in Louisianna

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Print the puzzles or type the solution on this excel file: 12 factors 843-852

My sister, Sue, lives in Louisiana. Several years ago Katrina upset her life, and now Harvey is pounding at her door. I have not heard from her since yesterday when she posted this dreary picture on facebook with the caption, “Flooded at my street.”

Sue, I hope you are okay. If you need a diversion, I hope this puzzle helps at least a tiny bit. I made it just for you. If you need someplace to stay, you can stay with me and my family. We send lots of love and prayers your way.

We also have a son, daughter-in-law, and two grandchildren who live in the Houston area. They are doing okay, but many of their friends are struggling. We pray for them as well.

After the freightening wind died down some, my daughter-in-law posted this picture with the caption, “Day 2 of Hurricane Harvey: We found a Craw-Dad in the back yard!”

My daughter-in-law later posted, “For those of you who are not in Houston I wanted to give you an update. We are located in Kingwood which is northeast of Houston. We have had rain since last Friday and many of our lakes, rivers and bayous are flowing out of their banks. Our home has been very blessed to be in a neighborhood where the rain water is draining nicely, so far. But many of our friends are not as lucky and have had to evacuate due to high water in their homes. We had one small leak in our kitchen, but were able to cover it and stop the dripping. We feel very blessed, but also very concerned for our friends and neighbors. Houston could use your prayers.”

I would like to add that Louisiana and several other towns and cities in Texas could use our prayers, help, and donations.

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Now I’ll write a little about the number 847:

844, 845, 846, 847, and 848 are the smallest five consecutive numbers whose square roots can be simplified.

847 is palindrome 1011101 in BASE 3 because 3⁶ + 3⁴ + 3³ + 3² + 3º = 847.

847 is also 700 in BASE 11 because 7(11²) = 847.

OEIS.org informs us that 847 is the sum of the digits of 2¹⁴ – 1, the 14th Mersenne prime. Since the sum of its digits is 847, that prime number has to be at least 95 digits long!

  • 847 is a composite number.
  • Prime factorization: 847 = 7× 11 × 11, which can be written 847 = 7 × 11²
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 847 has exactly 6 factors.
  • Factors of 847: 1, 7, 11, 77, 121, 847
  • Factor pairs: 847 = 1 × 847, 7 × 121, or 11 × 77
  • Taking the factor pair with the largest square number factor, we get √847 = (√121)(√7) = 11√7 ≈ 29.1032644

842 and Level 6

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Print the puzzles or type the solution on this excel file: 10-factors-835-842

29² + 1² = 842

That means 842 is the hypotenuse of a Pythagorean triple:

  • 58-840-842, calculated from 2(29)(1), 29² – 1², 29² + 1²

842 is repdigit 222 in BASE 20 because 2(20²) + 2(20¹) + 2(20º) = 842

  • 842 is a composite number.
  • Prime factorization: 842 = 2 × 421
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 842 has exactly 4 factors.
  • Factors of 842: 1, 2, 421, 842
  • Factor pairs: 842 = 1 × 842 or 2 × 421
  • 842 has no square factors that allow its square root to be simplified. √842 ≈ 29.017236