1793 Are You Easily Distracted?

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Today’s Puzzle:

Don’t let the x’s in the puzzle distract you! This puzzle can actually be solved quite easily! Just follow the previous suggestion of putting a 12 in one of the last two boxes, fill in the rest of the boxes (don’t worry if any of the numbers are greater than 12), identify the largest number, and adjust all of the numbers so that that largest number becomes the new 12.

Factors of 1793:

Solve this problem: 1 – 7 + 9 – 3 =

If the answer is 0 or any other multiple of 11, then 1793 is a multiple of 11.

  • 1793 is a composite number.
  • Prime factorization: 1793 = 11 × 163.
  • 1793 has no exponents greater than 1 in its prime factorization, so √1793 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1793 has exactly 4 factors.
  • The factors of 1793 are outlined with their factor pair partners in the graphic below.

More About the Number 1793:

1793 is a palindrome in base 32:
1O1 1(32²) + 24(32) + 1(1) = 1024 + 768 + 1 = 1793.
(O is the 15th letter of the alphabet, and 15 + 9 = 24, so O would be 24 if we all had 32 fingers.)

OEIS.org informs us that 1793 is a Fibonacci-inspired Pentanacci number.

 

Why Is 1792 a Friedman Number?

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Today’s Puzzle:

I’ve mentioned before that putting a 12 in one of the last two boxes will let you avoid negative numbers as you explore the relative relationship of the clues. For this puzzle, I would suggest that you put the 12 in the third from the last box. Why? Because the last triangle on the bottom has an 8 in it, and we will need to use either 12 – 8 = 4, and 4 – 2 = 2 for the last three boxes or 11 – 8 = 3, and 3 – 2 = 1.

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After you make your way to the empty triangle on the left of the puzzle, you will notice that you are missing the numbers 1 and 8. There isn’t any way to get a 5 by subtracting those two numbers, but if you realize that 13 – 5 = 8, you should know what adjustments you need to make to solve the puzzle.

Factors of 1792:

If the last digit of a number is 2 or 6, and the next-to-the-last digit is odd, then the whole number is divisible by 4.

If the last digit of a number is 0, 4, or 8, and the next-to-the-last digit is even, then the whole number is also divisible by 4.

1792 will allow us to apply those two divisibility observations several times as we make this factor tree:

  • 1792 is a composite number.
  • Prime factorization: 1792 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7, which can be written 1792 = 2⁸ × 7.
  • 1792 has at least one exponent greater than 1 in its prime factorization so √1792 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1792 = (√256)(√7) = 16√7.
  • The exponents in the prime factorization are 8 and 1. Adding one to each exponent and multiplying we get (8 + 1)(1 + 1) = 9 × 2 = 18. Therefore 1792 has exactly 18 factors.
  • The factors of 1792 are outlined with their factor pair partners in the graphic below.

More About the Number 1792:

1792 is a Friedman number because 7·2⁹⁻¹ = 1792.

Notice that the digits 1, 7, 9, and 2 and only those digits are used on both sides of the equal sign, and they are used the same number of times. 1792 is only the 26th Friedman number.

1792 is the difference of two squares in SEVEN different ways:
449² – 447² = 1792,
226² – 222² = 1792,
116² – 108² = 1792,
71² – 57² = 1792,
64² – 48² = 1792,
46² – 18² = 1792, and
44² – 12² = 1792.

1791 What a Distraction This Puzzle Is!

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Today’s Puzzle:

If you followed my advice from other posts and put a 12 in one of the last two boxes, you’ll be able to place five other numbers before hitting the roadblock that is the empty triangle.

Now we see that the highest known value is 15. The following numbers less than 15 are missing 1, 2, 3, 4, 5, 6, 8, 10, and 13. Since we have a 15, and our largest number can’t be greater than 12, let’s eliminate the smallest (15 – 12 = 3) three numbers from the list. We now have 4, 5, 6, 8, 10, and 13.

What can you do now? I suggest that you put an x such that -11 < x < 11 in the empty triangle and continue writing in values for the squares.

Regardless if x is a positive number or a negative number, the smallest number in a box will be either 7 or else 5 + x.

Since there isn’t a 6 + x or an 8, we know that one of those circled positions must be 1 and the other must be 2. If we assume the 7 should have been 2, we can lower the six numbers on the right of the puzzle by 5.

Then assuming that 5 + x must be 1 and filling in the puzzle we would get:

Uh oh! We can’t have two 9’s, 6’s, or 10’s, so those were NOT good assumptions.

I assure you that if switch the positions of the 1 and the 2, you will be able to complete the puzzle and place each number up to 12 in a box:

Factors of 1791:

  • 1791 is a composite number.
  • Prime factorization: 1791 = 3 × 3 × 199, which can be written 1791 = 3² × 199.
  • 1791 has at least one exponent greater than 1 in its prime factorization so √1791 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1791 = (√9)(√199) = 3√199.
  • The exponents in the prime factorization are 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1) = 3 × 2 = 6. Therefore 1791 has exactly 6 factors.
  • The factors of 1791 are outlined with their factor pair partners in the graphic below.

More About the Number 1791:

1791 is the difference of two squares in three different ways:
896² – 895² = 1791,
300² – 297² = 1791, and
104² – 95² = 1791.

1791 is A7A in base 13 because 10(13²) + 7(13) + 10(1) = 1791, and
636 in base 17 because 6(17²) + 3(17) + 6(1) = 1791.

1790 How Can You Solve This Subtraction Distraction?

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Today’s Puzzle:

How would I solve this puzzle? I would want to find a set of 12 consecutive numbers that were all positive and relatively small. Since there is an empty triangle near the right side of the puzzle, I would begin with the triangle on the bottom with a 4 in it and write 12 in a box above it. Then I would think and write 12 – 4 = 8 for the other box above the 4. My thinking would look like this:

  • 12
  • 12 – 4 = 8 (going to the right of the 12)
  • 12 – 6 = 6 (going to the left of the 12)
  • 6 + 7 = 13
  • 13 – 6 = 7
  • 7 – 2 = 5
  • 5 + 5 = 10
  • 10 + 5 = 15
  • 15 – 6 = 9

So that the puzzle looks like this:

I would note that I’m missing the following numbers: 1, 2, 3, 4, 11, and 14, and would figure out which of those missing numbers fit in the last three squares. Because I have a 15, I would note that 15 – 12 = 3 and would subtract 3 from each square to get numbers from 1 to 12. Figuring out what belongs in the empty triangle won’t be difficult either.

Factors of 1790:

  • 1790 is a composite number.
  • Prime factorization: 1790 = 2 × 5 × 179.
  • 1790 has no exponents greater than 1 in its prime factorization, so √1790 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1790 has exactly 8 factors.
  • The factors of 1790 are outlined with their factor pair partners in the graphic below.

More About the Number 1790:

1790 is the hypotenuse of a Pythagorean triple:
1074-1432-1790 which is (3-4-5) times 358.

1789 is 414 in base 21, but
1790 is 4I4 in base 19 because 4(19²) + 18(19) + 4(1) = 1790.

1789 An Easy Way to Solve This Subtraction Distraction Puzzle

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Today’s Puzzle:

To begin, I want to find a set of twelve consecutive numbers that will make the puzzle work. I want all of those numbers to be positive and relatively small. Thus, I would want to put a 12 in one of the last two boxes. Since the last triangle is -9, I would put the 12 in the last box. (If it were +9, I would put the 12 in the next to the last box.) Then I would do the following calculations based on the numbers in the triangles from right to left:

  • 12
  • 12 – 9 = 3
  • 3 + 4 = 7
  • 7 – 3 = 4
  • 4 + 4 = 8
  • 8 – 2 = 6
  • 6 – 1 = 5
  • 5 + 8 = 13
  • 13 – 4 = 9

And I would put the answers in the boxes from right to left:

The empty triangle makes me have to stop. Now I know I have to make some adjustments because one of the boxes has a 13 in it, but how much do I need to adjust each of those numbers? To answer that question, I will note what numbers from 1 to 12 are missing. I am missing 1, 2, 10, 11. The 13 I have means I can’t have the 1. I next access which of those missing numbers will yield -1. I note that 10 – 11 = -1, and write those numbers above the -1 triangle.

That leaves only the number 2 to place, but 10 + 4 ≠ 2, but 14. I place the 14 instead of the 2.

Now I have the twelve consecutive numbers from 3 to 14 in the boxes. If I subtract 2 from each of those twelve numbers, I will have all the numbers from 1 to 12. Also, it is easy to see that the number missing from the empty triangle is 2 whether I use 11 – 9 or 9 – 7.

Factors of 1789:

  • 1789 is a prime number.
  • Prime factorization: 1789 is prime.
  • 1789 has no exponents greater than 1 in its prime factorization, so √1789 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1789 has exactly 2 factors.
  • The factors of 1789 are outlined with their factor pair partners in the graphic below.

How do we know that 1789 is a prime number? If 1789 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1789. Since 1789 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1789 is a prime number.

More About the Number 1789:

1789 is the sum of two squares:
42² + 5² = 1789.

1789 is the hypotenuse of a Pythagorean triple:
420-1739-1789 calculated from 2(42)(5), 42² – 5², 42² + 5².

Here’s another way we know that 1789 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 42² + 5² = 1789 with 42 and 5 having no common prime factors, 1789 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1789. Since 1789 is not divisible by 5, 13, 17, 29, 37, or 41, we know that 1789 is a prime number.

1789 is in the twin prime 1787, 1789.

1789 is in the prime triplet 1783, 1787, 1789.

The first four multiples of 1789 are 1789, 3578, 5367, and 7156. Each of those multiples contains a 7. OEIS.org informs us that 1789 is the smallest number that can make that claim.

1789 is 414 in base 21 because 4(21²) + 1(21) + 4(1) = 1789.

 

1788 Subtraction Distraction

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Today’s Puzzle:

Factors of 1788:

A factor tree for 1788 isn’t very big because one of its prime factors has 3 digits.

  • 1788 is a composite number.
  • Prime factorization: 1788 = 2 × 2 × 3 × 149, which can be written 1788 = 2² × 3 × 149.
  • 1788 has at least one exponent greater than 1 in its prime factorization so √1788 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1788 = (√4)(√447) = 2√447.
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each exponent and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1788 has exactly 12 factors.
  • The factors of 1788 are outlined with their factor pair partners in the graphic below.

More About the Number 1788:

1788 is the hypotenuse of a Pythagorean triple:
612-1680-1788 which is 12 times (51-140-149).

1788 is the difference of two squares in two different ways:
448² – 446² = 1788, and
152² – 146² = 1788.

Two more square facts about 1788:
227² – 226² + 225² – 224² + 223² – 222² + 221² – 220² = 1788.

86² – 85² + 84² – 83² + 82² – 81² + 80² – 79² + 78² – 77² + 76² – 75² + 74² – 73² + 72² – 71² + 70² – 69² + 68² – 67² + 66² – 65² + 64² – 63² = 1788.

1787 The11-Digit Palindromes of Base 2

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Today’s Puzzle:

1787 is an 11-digit palindrome in base 2. I wondered how many 11-digit palindromes there are, what they are, and what numbers they represent in base ten. I decided to try to make you wonder about all that as well. Try it out yourself before you read how I solved this puzzle.

The only digits in base 2, are 0 and 1. The first digit of any number must be 1 or else the number will not have eleven digits. The last digit also must be one for the number to be a palindrome. In fact, all five last digits will be determined by the first five digits. Thus, we only need to find all possible combinations of 0 and 1 that can occur in the second through sixth positions. There are 2⁵ ways to write 0 and 1 in those 5 positions. That means we know right away that there are 32 different 11-digit palindromes in base 2. I opened Excel and wrote those 32 different 11-digit numbers beginning with 00000 and ending with 11111. I put a 1 in front of them and had Excel copy the appropriate numbers into the last 5 spots. That gave me all the 11-digit palindromes. Then I had Excel multiply the values in each cell with the powers of 2 that head up each column to give the base 10 representations. This chart was the final product.

Did you notice that the first base 10 number in the chart is the number just after 2¹º and the last number is the number right before 2¹¹?

Factors of 1787:

  • 1787 is a prime number.
  • Prime factorization: 1787 is prime.
  • 1787 has no exponents greater than 1 in its prime factorization, so √1787 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1787 has exactly 2 factors.
  • The factors of 1787 are outlined with their factor pair partners in the graphic below.

How do we know that 1787 is a prime number? If 1787 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1787. Since 1787 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, or 41, we know that 1787 is a prime number.

More About the Number 1787:

1787 and 1789 are twin primes.
1783, 1787, and 1789 are a prime triplet.

1787 is a palindrome in some other bases, too!
It’s 919 in base 14 because 9(14²) + 1(14) + 9(1) = 1787,
595 in base 18 because 5(18²) + 9(18) + 5(1) = 1787, and
191 in base 38 because 1(38²) + 9(38) + 1(1) = 1787.

1786 is a Centered Triangular Number

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Today’s Puzzle:

A formula for the nth triangular number is n(n+1)/2. Centered triangular numbers are the sum of three consecutive triangular numbers. What would be a formula for finding centered triangular numbers? What value of n in your formula would produce the number 1786?

Factors of 1786:

  • 1786 is a composite number.
  • Prime factorization: 1786 = 2 × 19 × 47.
  • 1786 has no exponents greater than 1 in its prime factorization, so √1786 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1786 has exactly 8 factors.
  • The factors of 1786 are outlined with their factor pair partners in the graphic below.

More About the Number 1786:

From OEIS.org we learn that 1786³ = 5,696,975,656. Notice that all those digits are 5 or greater.

1786 is 1G1 in base35,
because 1(35²) + 16(35) + 1(1) = 1786.

 

1785 A Pythagorean Triple Logic Puzzle

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Today’s Puzzle:

If you can print a copy of the puzzle from this Excel sheet, 10 Factors1773-1785, it will look like this:

Note: I have revised this puzzle since originally publishing it. I was horrified to discover that the original puzzle had two solutions. I apologize for any inconvenience I may have caused. This revised puzzle only has one solution.

Factors of 1785:

17 × 5 = 85, so 1785 is divisible by 17.

  • 1785 is a composite number.
  • Prime factorization: 1785 = 3 × 5 × 7 × 17.
  • 1785 has no exponents greater than 1 in its prime factorization, so √1785 cannot be simplified.
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1785 has exactly 16 factors.
  • The factors of 1785 are outlined with their factor pair partners in the graphic below.

More About the Number 1785:

Did you notice that 3, 5, 7, and 357 are all factors of 1785?
Or that 35 and 51 make a factor pair, and 3, 5, and 1 are also factors?

1785 is the hypotenuse of FOUR Pythagorean triples:
273-1764-1785
756-1617-1785
840-1575-1785
1071-1428-1785

1785 is the difference of two squares in EIGHT different ways:
893² – 892² = 1785,
299² – 296² = 1785,
181² – 176² = 1785, and five more ways. Can you find them?

1785 is a Palindrome in a couple of bases:
It’s 123321 in base 4, because 1(1024) + 2(256) + 3(64) + 3(16) + 2(4) + 1(1) = 1785.
And it’s 3F3 in base 22, because 3(22²) + 15(22) + 3(1) = 1785.

1784 Another Hundred Simplifiable Square Roots

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Today’s Puzzle:

What percentage of natural numbers less than or equal to 1784 have simplifiable square roots?

Here is a chart of the 601st to the 700th simplifiable square roots:

You can figure out the percentage of numbers up to 1784 that have simplifiable square roots by calculating 700×100 ÷1784.

Was the percentage higher or lower than you expected?

The green areas on the chart are for consecutive numbers with simplifiable square roots. 1680-1684 are the smallest five consecutive numbers that can make that claim. Why can they? Because every one of their prime factorizations has an exponent greater than one in it.

1680 prime factorization

Factors of 1784:

  • 1784 is a composite number.
  • Prime factorization: 1784 = 2 × 2 × 2 × 167, which can be written 1784 = 2³ × 167.
  • 1784 has at least one exponent greater than 1 in its prime factorization so √1784 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1784 = (√4)(√446) = 2√446.
  • 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 1784 has exactly 8 factors.
  • The factors of 1784 are outlined with their factor pairs in the graphic below.

More About the Number 1784:

1784 is the difference of two squares in two different ways:
447² – 445² = 1784, and
225² – 221² = 1784.

1784 is a palindrome in two bases:
It’s 494 in base20 because 4(20²)+9(20)+4(1) = 1784, and
2C2 in base27 because 2(27²)+12(27)+2(1) = 1784.