1551 Two Straight Lines

Today’s Puzzle:

The clues in this level 4 puzzle form two straight lines, and most of the logic needed to solve it is rather straightforward. Can you find the factors from 1 to 10 without getting twisted up?

Factors of 1551:

1+5 = 6, a multiple of 3, so 1551 is divisible by 3.
Since 1551 is a palindrome with an even number of digits, it is also divisible by 11.

  • 1551 is a composite number.
  • Prime factorization: 1551 = 3 × 11 × 47.
  • 1551 has no exponents greater than 1 in its prime factorization, so √1551 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 1551 has exactly 8 factors.
  • The factors of 1551 are outlined with their factor pair partners in the graphic below.

A Little More about the Number 1551:

Did you notice how many of 1551’s factors are palindromes?

1551 is the difference of two squares in four different ways:
776² – 775² = 1551,
260² – 257² = 1551,
76² – 65² = 1551, and
40² – 7² = 1551. (Yes, we are just 7² or 49 numbers away from 1600, the next perfect square!)

1529 Pointy Hat

Today’s Puzzle:

A pointy hat is part of many different Halloween costumes: Wizards, Witches, Medieval Princesses, and Clowns come to my mind. Today’s puzzle looks like a pointy hat. Use logic to work your magic in solving it! As always, there is only one solution.

Factors of 1529:

Look at this math fact using the digits of 1529:
1 – 5 + 2 – 9 = -11, a number divisible by 11, so 1529 is divisible by 11.

  • 1529 is a composite number.
  • Prime factorization: 1529 = 11 × 139.
  • 1529 has no exponents greater than 1 in its prime factorization, so √1529 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 1529 has exactly 4 factors.
  • The factors of 1529 are outlined with their factor pair partners in the graphic below.

More about the Number 1529:

1529 is the difference of two squares in two different ways:
765² – 764²  = 1529, and
75² – 64²  = 1529.

 

1516 and Level 4

Today’s Puzzle:

Using logic, write each number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues.

Factors of 1516:

  • 1516 is a composite number.
  • Prime factorization: 1516 = 2 × 2 × 379, which can be written 1516 = 2² × 379
  • 1516 has at least one exponent greater than 1 in its prime factorization so √1516 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1516 = (√4)(√379) = 2√379
  • 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 1516 has exactly 6 factors.
  • The factors of 1516 are outlined with their factor pair partners in the graphic below.

One More Fact about the Number 1516:

Only one of its factor pairs add up to an even number, so 1516 is the difference of two squares in only one way:
380² – 378² = 1516.

1493 and Level 4

Today’s Puzzle:

Write the numbers from 1 to 12 in both the first column and the top row so that those numbers are the factors of the clues given in the puzzle:

Factors of 1493:

  • 1493 is a prime number.
  • Prime factorization: 1493 is prime.
  • 1493 has no exponents greater than 1 in its prime factorization, so √1493 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1493 has exactly 2 factors.
  • The factors of 1493 are outlined with their factor pair partners in the graphic below.

How do we know that 1493 is a prime number? If 1493 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1493. Since 1493 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1493 is a prime number.

Other Facts about the number 1493:

1493 is the last prime number in the fourth prime quintuplet,
(1481, 1483, 1487, 1489, 1493), which is the smallest prime quintuplet that is not also part of a prime sextuplet.
In prime quintuplets, the first three numbers, the middle three numbers, and the last three numbers each form a prime triplet. Thus,1493 is the last prime number in the third prime triplet formed from the numbers in the fourth prime quintuplet.

1493 is the sum of two squares:
38² + 7² = 1493

1493 is the hypotenuse of a Pythagorean triple:
532-1395-1493, calculated from 2(38)(7), 38² – 7², 38² + 7²

Here’s another way we know that 1493 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 38² + 7² = 1493 with 38 and 7 having no common prime factors, 1493 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1493. Since 1493 is not divisible by 5, 13, 17, 29, or 37, we know that 1493 is a prime number.

1472 and Level 4

Today’s Puzzle:

Can you use logic to figure out where the factors from 1 to 12 must go to make the given clues be the products of those factors?

Factors of 1472:

  • 1472 is a composite number.
  • Prime factorization: 1472 = 2 × 2 × 2 × 2 × 2 × 2 × 23, which can be written 1472 = 2⁶ × 23.
  • 1472 has at least one exponent greater than 1 in its prime factorization so √1472 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1472 = (√64)(√23) = 8√23
  • The exponents in the prime factorization are 6 and 1. Adding one to each exponent and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1472 has exactly 14 factors.
  • The factors of 1472 are outlined with their factor pair partners in the graphic below.

A Little More about the Number 1472:

1472 is the difference of two squares in FIVE different ways:
369² – 367² = 1472
186² – 182² = 1472
96² – 88² = 1472
54² – 38² = 1472
39² – 7² = 1472

 

1459 and Level 4

Today’s Puzzle:

Can you place the numbers 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues? You might surprise yourself with how well you do!

Factors of 1459:

  • 1459 is a prime number.
  • Prime factorization: 1459 is prime.
  • 1459 has no exponents greater than 1 in its prime factorization, so √1459 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1459 has exactly 2 factors.
  • The factors of 1459 are outlined with their factor pair partners in the graphic below.

How do we know that 1459 is a prime number? If 1459 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1459. Since 1459 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1459 is a prime number.

Other Facts about the number 1459:

1459 is the difference of two squares:
730² – 729² = 1459

From OEIS.org we learn this chilling cubic fact:
1³ + 4³ + 5³ + 9³ = 919, and
9³ + 1³ + 9³ = 1459

 

1446 Peppermint Stick

Red and green striped peppermint sticks are often seen in stores and homes in December. Can you lick this peppermint stick puzzle or will you let it lick you?

The puzzle number was 1446. Here are a few facts about that number:

  • 1446 is a composite number.
  • Prime factorization: 1446 = 2 × 3 × 241
  • 1446 has no exponents greater than 1 in its prime factorization, so √1446 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 1446 has exactly 8 factors.
  • The factors of 1446 are outlined with their factor pair partners in the graphic below.

1446 is also the hypotenuse of a Pythagorean triple:
720-1254-1446 which is 6 times (120-209-241)

1435 is the 23rd Friedman Number

41 ×35 = 1435. Since the same digits are used on both sides of the equal sign that makes 1435 the 23rd Friedman number.

Since it’s good for puzzles to have a puzzle number to distinguish them from each other, I’ve given this one the number 1435. Can you solve it?

Print the puzzles or type the solution in this excel file:  10 Factors 1432-1442

Here are a few more facts about the number 1435:

  • 1435 is a composite number.
  • Prime factorization: 1435 = 5 × 7 × 41
  • 1435 has no exponents greater than 1 in its prime factorization, so √1435 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 1435 has exactly 8 factors.
  • The factors of 1435 are outlined with their factor pair partners in the graphic below.

1435 is also the hypotenuse of FOUR Pythagorean triples:
315-1400-1435 which is 35 times (9-40-41)
588-1309-1435 which is 7 times (84-187-205)
861-1148-1435 which is (3-4-5) times 287
931-1092-1435 which is 7 times (133-156-205)

1413 and Level 4

You find a rhythm as you solve this level 4 puzzle. The logic is quite easy for most of it, but there is at least one place that will require you to think things through before proceeding.

Print the puzzles or type the solution in this excel file: 10 Factors 1410-1418

Now I’ll share a few facts about the puzzle number, 1413:

  • 1413 is a composite number.
  • Prime factorization: 1413 = 3 × 3 × 157, which can be written 1413 = 3² × 157
  • 1413 has at least one exponent greater than 1 in its prime factorization so √1413 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1413 = (√9)(√157) = 3√157
  • 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 1413 has exactly 6 factors.
  • The factors of 1413 are outlined with their factor pair partners in the graphic below.

1413 is the sum of two squares:
33² + 18² = 1413

That means that 1413 is the hypotenuse of a Pythagorean triple:
765-1188-1413 calculated from 33² – 18², 2(33)(18), 33² + 18².
It is also 9 times (85-132-157)

1397 and Level 4

I bet you know enough multiplication facts to get this puzzle started. Once you’ve started it, you might as well finish it. You will feel so clever when you do!

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

Now I’ll write a little bit about the puzzle number, 1397:

  • 1397 is a composite number.
  • Prime factorization: 1397 = 11 × 127
  • 1397 has no exponents greater than 1 in its prime factorization, so √1397 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 1397 has exactly 4 factors.
  • The factors of 1397 are outlined with their factor pair partners in the graphic below.

1397 is the difference of two squares two different ways:
699² – 698² = 1397
69² – 58² = 1397