Puzzles

848 and Level 3

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

848 is a palindrome, and all but three of its factors are palindromes, too. (Single digit numbers are also palindromes.)

848 is the sum of two squares: 28² + 8² = 848

848 is the hypotenuse of a Pythagorean triple:

  • 448-720-848, calculated from 2(28)(8), 28² – 8², 28² + 8²

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

  • 848 is a composite number.
  • Prime factorization: 848 = 2 × 2 × 2 × 2 × 53, which can be written 848 = 2⁴ × 53
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 848 has exactly 10 factors.
  • Factors of 848: 1, 2, 4, 8, 16, 53, 106, 212, 424, 848
  • Factor pairs: 848 = 1 × 848, 2 × 424, 4 × 212, 8 × 106, or 16 × 53
  • Taking the factor pair with the largest square number factor, we get √848 = (√16)(√53) = 4√53 ≈ 29.1204396

847 Sending Love to My Sister in Louisianna

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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

846 and Level 2

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

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

846 can be written as the sum of consecutive prime numbers two different ways. Together they use ALL the prime numbers from 13 to 127 exactly one time.

  • 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 846; that’s eighteen consecutive primes.
  • 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 846; that’s eight consecutive primes.

OEIS.org informs us that 846² = 715,716. Not quite a Ruth Aaron pair, but still quite impressive.

  • 846 is a composite number.
  • Prime factorization: 846 = 2 × 3 × 3 × 47, which can be written 846 = 2 × 3² × 47
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 846 has exactly 12 factors.
  • Factors of 846: 1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846
  • Factor pairs: 846 = 1 × 846, 2 × 423, 3 × 282, 6 × 141, 9 × 94, or 18 × 47
  • Taking the factor pair with the largest square number factor, we get √846 = (√9)(√94) = 2√209 ≈ 29.086079

843 is a Lucas Number

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843 is the hypotenuse of Pythagorean triple 480-693-843 which is 3 times (160-231-281).

I think 843 looks interesting in a couple different bases:

  • It is palindrome 1101001011 in BASE 2 because 2⁹ + 2⁸ + 2⁶ + 2³ + 2¹ + 2⁰ = 843
  • It is 123 in BASE 28 because 1(28²) + 2(28¹) + 3(28º) = 843

From OEIS.org, I learned that 843 is the 14th Lucas number which is similar to being a Fibonacci number.

  • Fibonacci numbers start with 1, 1, and all the rest of the Fibonacci numbers are found by adding the previous two numbers in the series.
  • Lucas numbers start with 2, 1, and all the rest of the Lucas numbers are found by adding the previous two numbers in the series.

843 makes the list of Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, . . .

Even though it is the 15th number on the list, it is the 14th Lucas number. (The list starts with the zeroth number?)

Here is 843’s factoring information:

  • 843 is a composite number.
  • Prime factorization: 843 = 3 × 281
  • 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 843 has exactly 4 factors.
  • Factors of 843: 1, 3, 281, 843
  • Factor pairs: 843 = 1 × 843 or 3 × 281
  • 843 has no square factors that allow its square root to be simplified. √843 ≈ 03446228

 

 

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

 

839 and Level 5

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839 is the sum of the five prime numbers from 157 to 179:

  • 157 + 163 + 167 + 173 + 179 = 839

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

  • 839 is a prime number.
  • Prime factorization: 839 is prime.
  • The exponent of prime number 839 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 839 has exactly 2 factors.
  • Factors of 839: 1, 839
  • Factor pairs: 839 = 1 × 839
  • 839 has no square factors that allow its square root to be simplified. √839 ≈ 28.9654967

How do we know that 839 is a prime number? If 839 were not a prime number, then it would be divisible by at least one prime number less than or equal to √839 ≈ 28.97. Since 839 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 839 is a prime number.

 

In Which Bases is 838 a Palindrome?

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838 is a palindrome in base 10. Is it a palindrome in any other bases? Yes, two others.

  • 262 BASE 19 because 2(19²) + 6(19¹) + 2(19º) = 838
  • 141 BASE 27 because 1(27²) + 4(27¹) + 1(27º) = 838

There is only one way 838 can be written as the sum of consecutive numbers:

  • 208 + 209 + 210 + 211 = 838

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

  • 838 is a composite number.
  • Prime factorization: 838 = 2 × 419
  • 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 838 has exactly 4 factors.
  • Factors of 838: 1, 2, 419, 838
  • Factor pairs: 838 = 1 × 838 or 2 × 419
  • 838 has no square factors that allow its square root to be simplified. √838 ≈ 28.9482

 

 

837 and Level 3

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837 is repdigit RR in BASE 30 (R is 27 in base 10). All that means is that 27(30¹) + 27(30º) = 837, which naturally follows from the fact that 27(30 + 1) = 837.

837 has four odd factor pairs, so 837 can be written as the difference of two squares four different ways:

  • 837 × 1 = 837 means 419² – 418² = 837
  • 279 × 3 = 837 means 141² – 138² = 837
  • 93 × 9 = 837 means 51² – 42² = 837
  • 31 × 27 = 837 means 29² – 2² = 837

Hmm…837 is only four numbers away from the next perfect square, 841.

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

  • 837 is a composite number.
  • Prime factorization: 837 = 3 × 3 × 3 × 31, which can be written 837 = 3³ × 31
  • 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 837 has exactly 8 factors.
  • Factors of 837: 1, 3, 9, 27, 31, 93, 279, 837
  • Factor pairs: 837 = 1 × 837, 3 × 279, 9 × 93, or 27 × 31
  • Taking the factor pair with the largest square number factor, we get √837 = (√9)(√93) = 3√93 ≈ 28.93095

 

 

Finding Ways to Write 836 as the Sum of Consecutive Numbers

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OEIS.org informs us that 836² = 698,896, a palindrome.

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

836 can be written as the sum of 11 consecutive numbers and as the sum of 19 consecutive numbers because 11 and 19 are its odd factors (not including 1) that are less than 41. (Remember 861 is the 41st triangular number.) Notice 836’s factor pairs highlighted in red.

  • 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 + 81 = 836; that’s 11 consecutive numbers
  • 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 836; that’s 19 consecutive numbers

836 can be written as the sum of 8 consecutive numbers. Why? Because its factor that is the greatest power of 2 is 4, and because 1 is a factor of 836. Note that 2(4)(1) = 8.

  • 101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 = 836

836 can’t be written as the sum of 2(4)(11) = 88 consecutive numbers or 2(4)(19) = 152 consecutive numbers because neither 88 or 152 is less than 41.

  • 836 is a composite number.
  • Prime factorization: 836 = 2 × 2 × 5 × 41, which can be written 836 = 2² × 11 × 19
  • 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 836 has exactly 12 factors.
  • Factors of 836: 1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836
  • Factor pairs: 836 = 1 × 836, 2 × 418, 4 × 209, 11 × 76, 19 × 44, or 22 × 38
  • Taking the factor pair with the largest square number factor, we get √836 = (√4)(√209) = 2√209 ≈ 28.91366

835 and Level 1

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835 is the hypotenuse of a Pythagorean triple, 501-668-835, which is 167 times (3-4-5).

835 can be written as the difference of two squares two different ways:

  • 418² – 417² = 835
  • 86² – 81² = 835

835 can be written as the sum of consecutive numbers three different ways.

  • 417 + 418 = 835; that’s two consecutive numbers.
  • 165 + 166 + 167 + 168 + 169 = 835; that’s five consecutive numbers.
  • 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 + 87 + 88 = 835; that’s ten consecutive numbers.

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

  • 835 is a composite number.
  • Prime factorization: 835 = 5 × 167
  • 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 835 has exactly 4 factors.
  • Factors of 835: 1, 5, 167, 835
  • Factor pairs: 835 = 1 × 835 or 5 × 167
  • 835 has no square factors that allow its square root to be simplified. √835 ≈ 28.89636655