Level 1 Puzzle

876 and Level 1

/ ivasallay / Leave a comment

876 consists of three consecutive numbers 6, 7, 8, so 876 has to be divisible by 3. We can also conclude the following:

  • Since it’s even and divisible by 3, we know that 876 is also divisible by 6.
  • Since it is divisible by 3 and it’s last two digits are divisible by 4, we know that 876 is also divisible by 12.

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

876 is a palindrome in four other bases:

  • 727 BASE 11, because 7(121) + 2(11) + 7(1) = 876
  • 525 BASE 13, because 5(13²) + 2(13¹) + 5(13º) = 876
  • 282 BASE 19, because 2(19²) + 8(19¹) + 2(19º) = 876
  • 1A1 BASE 25 (A is 10 base 10), because 1(25²) + 10(25¹) + 1(25º) = 876

876 is also the hypotenuse of Pythagorean triple, 576-660-876 which is 12 times (48-55-73).

  • 876 is a composite number.
  • Prime factorization: 876 = 2 × 2 × 3 × 73, which can be written 876 = 2² × 3 × 73
  • 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 876 has exactly 12 factors.
  • Factors of 876: 1, 2, 3, 4, 6, 12, 73, 146, 219, 292, 438, 876
  • Factor pairs: 876 = 1 × 876, 2 × 438, 3 × 292, 4 × 219, 6 × 146, or 12 × 73,
  • Taking the factor pair with the largest square number factor, we get √876 = (√4)(√219) = 2√219 ≈ 29.597297

865 and Level 1

/ ivasallay / 2 Comments

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

865 is the sum of two squares two different ways:

  • 28² + 9² = 865
  • 24² + 17² = 865

865 is the hypotenuse of four Pythagorean triples, two of which are primitives:

  • 260-825-865, which is 5 times (52-165-173)
  • 287-816-865, which is 24² – 17², 2(24)(17), 24² + 17²
  • 504-703-865 which is 2(28)(9), 28² – 9², 28² – 9²
  • 519-692-865, which is (3-4-5) times 173

You could see 865’s factors in two of those Pythagorean triples, and here they are again:

  • 865 is a composite number.
  • Prime factorization: 865 = 5 × 173
  • 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 865 has exactly 4 factors.
  • Factors of 865: 1, 5, 173, 865
  • Factor pairs: 865 = 1 × 865 or 5 × 173
  • 865 has no square factors that allow its square root to be simplified. √865 ≈ 29.41088

853 You Can Do This Puzzle!

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This is a level 1 puzzle that is easier than even most other level 1 puzzles. You can do this puzzle! If mathematics makes you uncomfortable, you can still do this puzzle! Even if math class is your worse nightmare, you can complete this puzzle, and gain a little confidence. Go ahead, give it a try! Figure out where each number from one to ten goes in the top row and also in the first column so that the puzzle turns into a mixed-up multiplication table. It’s easier and far less time consuming than Sudoku. You CAN do this puzzle! Then, after you find all the factors, and are feeling really good about yourself, IF you want, you can fill in all the other cells of this mixed up multiplication table.

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

853 is a prime number that leaves a remainder of 1 when divided by 4, so 853 is the hypotenuse of a Pythagorean triple: 205-828-853.

23² + 18² = 853 so 205-828-853 can be calculated from 23² – 18², 2(23)(18), 23² + 18².

  • 853 is a prime number.
  • Prime factorization: 853 is prime.
  • The exponent of prime number 853 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 853 has exactly 2 factors.
  • Factors of 853: 1, 853
  • Factor pairs: 853 = 1 × 853
  • 853 has no square factors that allow its square root to be simplified. √853 ≈ 29.20616

How do we know that 853 is a prime number? If 853 were not a prime number, then it would be divisible by at least one prime number less than or equal to √853 ≈ 29.2. Since 853 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 853 is a prime number.

Here’s another way we know that 853 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 23² + 18² = 853 with 23 and 18 having no common prime factors, 853 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √853 ≈ 29.2. Since 853 is not divisible by 5, 13, 17, or 29, we know that 853 is a prime number.

 

843 is a Lucas Number

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

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

 

 

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

822 and Level 1

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The sum of 822’s digits is 12, a number divisible by 3. That means that even number 822 can be evenly divided by 2, 3, and 6.

822 is the sum of the 12 prime numbers from 43 to 97.

822 is palindrome 212 in base 20 because 2(20²) + 1(20¹) + 2(20º) = 822.

Print the puzzles or type the solution on this excel file: 10-factors-822-828

  • 822 is a composite number.
  • Prime factorization: 822 = 2 x 3 x 137
  • 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 822 has exactly 8 factors.
  • Factors of 822: 1, 2, 3, 6, 137, 274, 411, 822
  • Factor pairs: 822 = 1 x 822, 2 x 411, 3 x 274, or 6 x 137
  • 822 has no square factors that allow its square root to be simplified. √822 ≈ 28.67054

815 and Level 1

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Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

  • 489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

 

Print the puzzles or type the solution on this excel file: 12 factors 815-820

  • 815 is a composite number.
  • Prime factorization: 815 = 5 x 163
  • 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 815 has exactly 4 factors.
  • Factors of 815: 1, 5, 163, 815
  • Factor pairs: 815 = 1 x 815 or 5 x 163
  • 815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

807 and Level 1

/ ivasallay / 1 Comment

What can I say about the number 807?

807 is palindrome 151 in BASE 26 because 1(26²) + 5(26) + 1(1) = 807.

Anything else? Well, I can figure out a few other things because 807’s has two prime factors, 3 and 269:

We can write ANY number (unless it’s a power of 2) as the sum of consecutive numbers in at least one way. 807 has three different ways to do that:

  • 403 + 404 = 807 because 807 isn’t divisible by 2.
  • 268 + 269 + 270 = 807 because it is divisible by 3.
  • 132 + 133 + 134 + 135 + 136 + 137 = 807 since it is divisible by 3 but not by 6.

I know that one of 807’s factors, 269, is a hypotenuse of a Pythagorean triple, so 807 is also. Thus. . .

  • (3·69)² + (3·260)² = (3·269)², or in other words, 207² + 780² = 807²

Since 807 has two odd sets of factor pairs, I know that 807 can be written as the difference of two squares two different ways:

  • 136² – 133² = 807
  • 404² – 403² = 807

I don’t usually do this, but today’s puzzle has something in common with 807. Can you tell what it is?

Print the puzzles or type the solution on this excel file: 10-factors 807-814

  • 807 is a composite number.
  • Prime factorization: 807 = 3 x 269
  • 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 807 has exactly 4 factors.
  • Factors of 807: 1, 3, 269, 807
  • Factor pairs: 807 = 1 x 807 or 3 x 269
  • 807 has no square factors that allow its square root to be simplified. √807 ≈ 28.4077454

 

 

 

 

 

801 and Level 1

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When it comes to applying our tried and true trick for divisibility by nine to the number 801, zero is just a place holder. Thus, since 81 is divisible by 9, so is 801. Adding up its digits was hardly necessary.

  • 801 is a composite number.
  • Prime factorization: 801 = 3 x 3 x 89, which can be written 801 = (3^2) x 89
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 801 has exactly 6 factors.
  • Factors of 801: 1, 3, 9, 89, 267, 801
  • Factor pairs: 801 = 1 x 801, 3 x 267, or 9 x 89
  • Taking the factor pair with the largest square number factor, we get √801 = (√9)(√89) = 3√89 ≈ 28.301943396.

Would you be surprised to know the following division facts?

  • 81 ÷ 3 = 27
  • 801 ÷ 3 = 267
  • 8001 ÷ 3 = 2667
  • 80001 ÷ 3 = 26667 and so forth. The number of 6’s in the quotient is the same as the number of 0’s in the dividend!

Here are some more predictable division facts:

  • 81 ÷ 9 = 9
  • 801 ÷ 9 = 89
  • 8001 ÷ 9 = 889
  • 80001 ÷ 9 = 8889 and so forth. You guessed it! The number of 8’s in the quotient is the same as the number of 0’s in the dividend!

Even though you can’t see 81 in this puzzle with all perfect square clues, it isn’t difficult to see where 9 × 9 and 81 belong:

Print the puzzles or type the solution on this excel file: 10-factors 801-806

801 is a palindrome in three bases:

  • 1441 BASE 8 because 1(8^3) + 4(8^2) + 4(8) + 1(1) = 801
  • 2D2 BASE 17 D is 13 base 10 because 2(289) + 13(17) = 2(1) = 801
  • 171 BASE 25 because 1(25²) + 7(25) + 1(1) =801

801 is the sum of two squares:

  • 24² + 15² =801

So it follows that 801 is the hypotenuse of a Pythagorean triple:

  • 351-720-801 which is 9 times 39-80-89

801 is the sum of three squares TEN ways:

  1. 28² + 4² + 1² = 801
  2. 27² + 6² + 6² = 801
  3. 26² + 11² + 2² =801
  4. 26² + 10² + 5² = 801
  5. 24² + 12² + 9² = 801
  6. 23² + 16² + 4² = 801
  7. 22² + 14² + 11² = 801
  8. 21² + 18² + 6² = 801
  9. 20² + 20² + 1² = 801
  10. 17² + 16² + 16² = 801

OEIS.org gives us this last fun fact:

801 = (7! + 8! + 9! + 10!) / (7 × 8 × 9 × 10).

 

 

 

795 a Level 1 Puzzle with ♥

/ ivasallay / 2 Comments

I learned yesterday that pushing ALT on the computer key pad and 3 on the number pad at the same time produces a ♥ when the keys are released. I added a few ♥‘s to today’s puzzle because Valentine’s day is almost here:

795-puzzle

  • 795 is a composite number.
  • Prime factorization: 795 = 3 x 5 x 53
  • 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 795 has exactly 8 factors.
  • Factors of 795: 1, 3, 5, 15, 53, 159, 265, 795
  • Factor pairs: 795 = 1 x 795, 3 x 265, 5 x 159, or 15 x 53
  • 795 has no square factors that allow its square root to be simplified. √795 ≈ 28.195744.

795-factor-pairs

OEIS.org informs us that if we add up all the factors of 795 we get a number that is a perfect 4th power:

  • 1 + 3 + 5 + 15 + 53 + 159 + 265 + 795 = 1296, which happens to be the 4th power of 6. (Of course 1296 also equals 36 squared.)

795 is the hypotenuse of four Pythagorean triples and thus:

  • 69 – 792 – 795 which is 3 times (23 – 264 – 265)
  • 288 – 741 – 795 which is 3 times (96 – 247 – 265)
  • 420 – 675 – 795 which is 15 times (28 – 45 – 53)
  • 477 – 636 – 795 which is 159 times (3 – 4 – 5)

795 is also the sum of three squares two different ways:

  • 25² + 13² + 1² = 795
  • 25² + 11² + 7² = 795