1391 and Level 1

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Many of the clues in this puzzle have double digits. If you know why they do, then you can find all the factors and solve this puzzle!

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Here’s some information about the number 1391:

  • 1391 is a composite number.
  • Prime factorization: 1391 = 13 × 107
  • 1391 has no exponents greater than 1 in its prime factorization, so √1391 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 1391 has exactly 4 factors.
  • The factors of 1391 are outlined with their factor pair partners in the graphic below.

1391 is the hypotenuse of a Pythagorean triple:
535-1284-1391 which is (5-12-13) times 107

1390 Find the Factors (ax±b)(cx±d)

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I liked making a puzzle using trinomials earlier today. This one will take more skill to solve even though it contains fewer trinomials. Some of the factors will have negative numbers, and the leading coefficients of the trinomials are not 1.

In this puzzle, you can see the number 24 twice. It needs to be factored to solve the puzzle. It might be 3 × 8 or 4 × 6, but it can’t be 1 × 24 or 2 × 12 because for this puzzle ALL of the factors of 24 have to be non-zero integers from -10 to +10.

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Every factor must appear once in the first column and once in the top row. So if you put 2x + 5 in the top row, you will also have to put 2x + 5 somewhere in the first column as well.

Sometimes all of the terms in the trinomial have a common factor and can, therefore, be factored further, but don’t worry about that right now.

You will have to find all of the factors in the puzzle before you can figure out what the missing clue should be. That’s about all the mystery I can put in a puzzle like this. Good luck with it!

Since this is different than any other puzzle I’ve ever published, you can see the solution here:

Now I’ll share some information about the number 1390:

  • 1390 is a composite number.
  • Prime factorization: 1390 = 2 × 5 × 139
  • 1390 has no exponents greater than 1 in its prime factorization, so √1390 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 1390 has exactly 8 factors.
  • The factors of 1390 are outlined with their factor pair partners in the graphic below.

1390 is the hypotenuse of a Pythagorean triple:
834-1112-1390 which is (3-4-5) times 278

1390 is 102345 in BASE 6 making it the smallest number to use all the digits less than 6 in base 6. Thank you OEIS.org for that reminder.

1389 Positive Trinomial Puzzle

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Today on Twitter, Mr. Allen requested some good problem-solving resources for quadratics. He made up one himself.

I decided to make one as well. It is similar to my other Find the Factors puzzles. You will have to use logic to solve it, but in many ways, it will be easier to solve than most of my regular puzzles. Like always, there is only one solution.

Print the puzzles or type the solution in this excel file: 12 Factors 1389-1403

Every term is positive so if you already know how to factor trinomials it should be relatively easy to solve. All the factors from (x + 1) to (x + 9) need to appear exactly one time in both the first column and the top row of the puzzle.  Once all the factors are found, the puzzle is solved, but you can find all the products of those factors and write them in the body of the puzzle if you want.

Since this is my 1389th post, here’s a little bit about that number:

  • 1389 is a composite number.
  • Prime factorization: 1389 = 3 × 463
  • 1389 has no exponents greater than 1 in its prime factorization, so √1389 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 1389 has exactly 4 factors.
  • The factors of 1389 are outlined with their factor pair partners in the graphic below.

1389 is the difference of two squares in two different ways:
695² – 694² = 1389
233² – 230² = 1389

1388 Mystery Level

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Sometimes puzzles start out easy enough but get a little more complicated later on. Does that happen with this puzzle? There’s only one way to find out!


Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are some facts about the number 1388:

  • 1388 is a composite number.
  • Prime factorization: 1388 = 2 × 2 × 347, which can be written 1388 = 2² × 347
  • 1388 has at least one exponent greater than 1 in its prime factorization so √1388 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1388 = (√4)(√347) = 2√347
  • 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 1388 has exactly 6 factors.
  • The factors of 1388 are outlined with their factor pair partners in the graphic below.

1388 is the difference of two squares:
348² – 346² = 1388

The Shape of 1387 Tiny Squares

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22³ – 21³ = 1387, and that’s why it is the 22nd hexagonal number.

1387 is also the 19th decagonal number. Why? Because 4(19²) – 3(19) = 1387.

Here’s more about the number 1387:

  • 1387 is a composite number.
  • Prime factorization: 1387 = 19 × 73
  • 1387 has no exponents greater than 1 in its prime factorization, so √1387 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 1387 has exactly 4 factors.
  • The factors of 1387 are outlined with their factor pair partners in the graphic below.

 

1387 is the hypotenuse of a Pythagorean triple:
912-1045-1387 which is 19 times (48-55-73)

1386 What You Need to Know About the Multiplication Game

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Helpful Hints about the Multiplication Game:

I recently wrote about Hooda Math’s Multiplication Game. There’s a couple of things I didn’t tell you in that post.

First of all, you don’t have to use a computer to play the game. (However, using one the first time you play will help you understand how to play). You can print a game board to play. I’ve created a game board below that you could use. Each player can use different items such as beads, pennies, nickels, and dimes as markers to mark the factors used and to claim the resulting products on the game board.

The second thing you should know is that getting four squares in a row, horizontally, vertically or diagonally is NOT equally likely every place on the board. If one particular number is all you need to get a win, you are less likely to get that number if it only has one factor (like the numbers marked in yellow have). As far as this game is concerned, the products have the number of factors that I’ve indicated, even though in reality most of them have more than that.

You can’t win unless your opponent gives you one of the factors you need to claim that winning space. If 4 of the 9 possible factors will get it for you, the odds are much better your opponent will give you what you need than if only 1 of the 9 possible factors will do it.

If you know which numbers have four possible factors, you may have an advantage over someone who thinks this game is really just a variation of tic-tac-toe. Of course, those products with four factors could also make you more likely to get blocked as well! And if you use my colorful game board, your opponent will know just as much as you do about how many ways they can get each square.

1386 Factor Cake:

Since the biggest prime factor of 1386 is 11, it makes an especially festive factor cake!

Factors of 1386:

Now I’ll share some information about the number 1386:

  • 1386 is a composite number.
  • Prime factorization: 1386 = 2 × 3 × 3 × 7 × 11, which can be written 1386 = 2 × 3² × 7 × 11
  • 1386 has at least one exponent greater than 1 in its prime factorization so √1386 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1386 = (√9)(√154) = 3√154
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1386 has exactly 24 factors.
  • The factors of 1386 are outlined with their factor pair partners in the graphic below.

Sum-Difference Puzzle:

1386 has twelve factor pairs. One of the factor pairs adds up to 85, and a different one subtracts to 85. If you can identify those factor pairs, then you can solve this puzzle!

One More Fact about the Number 1386:

OEIS.org also noted that 1 + 3⁴ + 8 + 6⁴ = 1386.

1385 Mystery Level

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You can suspect that the common factor of 9 and 6 is either 1 or 3, but don’t jump to conclusions about which one will satisfy this mystery! There’s important evidence elsewhere in the puzzle that you should consider first.

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some facts about the puzzle number, 1385:

  • 1385 is a composite number.
  • Prime factorization: 1385 = 5 × 277
  • 1385 has no exponents greater than 1 in its prime factorization, so √1385 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 1385 has exactly 4 factors.
  • The factors of 1385 are outlined with their factor pair partners in the graphic below.

1385 is the sum of two squares in two different ways:
32² + 19² = 1385
37² + 4² = 1385

1385 is the hypotenuse of a Pythagorean triple:
296-1353-1385 calculated from 2(37)(4), 37² – 4², 37² + 4²
575-1260-1385 which is 5 times (115-252-277)
663-1216-1385 calculated from 32² – 19², 2(32)(19), 32² + 19²
831-1108-1385 which is (3-4-5) times 277

1384 and Level 6

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Guessing and checking can be so frustrating! If instead, you study the clues to find a logical place to start this puzzle, you are more likely to be able to find the one and only solution.

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the number 1384:

  • 1384 is a composite number.
  • Prime factorization: 1384 = 2 × 2 × 2 × 173, which can be written 1384 = 2³ × 173
  • 1384 has at least one exponent greater than 1 in its prime factorization so √1384 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1384 = (√4)(√346) = 2√346
  • The exponents in the prime factorization are 3 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1384 has exactly 8 factors.
  • The factors of 1384 are outlined with their factor pairs in the graphic below.

1384 is the hypotenuse of a Pythagorean triple:
416-1320-1384 which is 8 times (52-165-173)

1383 and Level 5

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Level 5 puzzles aren’t any harder than level 4 puzzles unless I trick you into starting with the common factor of a pair of clues that have more than one possibility. You won’t let me trick you, will you?

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Here are a few facts about the number 1383:

  • 1383 is a composite number.
  • Prime factorization: 1383 = 3 × 461
  • 1383 has no exponents greater than 1 in its prime factorization, so √1383 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 1383 has exactly 4 factors.
  • The factors of 1383 are outlined with their factor pair partners in the graphic below.

1383 is the hypotenuse of a Pythagorean triple:
783-1140-1383 which is 3 times (261-380-461)

1382 and Level 4

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After solving a couple of level 3 puzzles, you are ready to give a level 4 puzzle a try. It isn’t any more difficult to solve than a level 3, except that the clues are not given in a logical order. Don’t let that stop you from succeeding!

Print the puzzles or type the solution in this excel file: 10 Factors 1373-1388

Now I’ll share some information about the number 1382:

  • 1382 is a composite number.
  • Prime factorization: 1382 = 2 × 691
  • 1382 has no exponents greater than 1 in its prime factorization, so √1382 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 1382 has exactly 4 factors.
  • The factors of 1382 are outlined with their factor pair partners in the graphic below.

1382 is in one Pythagorean triple:
1382-477480-477482 calculated from 2(691)(1), 691² – 1², 691² + 1²