1625 is a Centered Square Number

Today’s Puzzle:

Because 1625 is the 29th centered square number, it is one more than four times the 28th triangular number. Can you draw lines on the graphic below separating out one tiny square and dividing the rest of the graphic into four equal triangles each with a base of 28 tiny squares?

Factors of 1625:

  • 1625 is a composite number.
  • Prime factorization: 1625 = 5 × 5 × 5 × 13, which can be written 1625 = 5³ × 13
  • 1625 has at least one exponent greater than 1 in its prime factorization so √1625 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1625 = (√25)(√65) = 5√
  • 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 1625 has exactly 8 factors.
  • The factors of 1625 are outlined with their factor pairs in the graphic below.

More about the Number 1625:

1625 is the sum of two squares FOUR different ways:
40² + 5² = 1625,
37² + 16² = 1625,
35² + 20² = 1625, and
29² + 28² = 1625.

1625 is the hypotenuse of TEN Pythagorean triples:
57-1624-1625, calculated from 29² – 28², 2(29)(28), 29² + 28²,
180-1615-1625, which is 5 times (36-323-325),
400-1575-1625, calculated from 2(40)(5), 40² – 5², 40² + 5²,
455-1560-1625, which is (7-24-25) times 65,
572-1521-1625, which is 13 times (44-117-125),
625-1500-1625, which is (5-12-13) times 125,
825-1400-1625, calculated from 35² – 20², 2(35)(20), 35² + 20²,
975-1300-1625, which is (3-4-5) times 325,
1020-1265-1625, which is 5 times (204-253-325), and
1113-1184-1625, calculated from 37² – 16², 2(37)(16), 37² + 16².

 

1624 Applying Twelve Divisibility Rules to Permutations of 1234567890

Today’s Puzzle:

Can you use divisibility rules to find a number that uses all ten digits exactly once and is divisible by all the numbers from 1 to 10?”

EVERY such number will be divisible by 1, of course. That’s the divisibility rule for 1. That one was super easy.

But guess what! No matter how you arrange those ten digits it will be divisible by 3 and by 9. Why? Because the sum of the digits of 1234567890 is 45, a number divisible by both 3 and 9. That’s the divisibility rule for 3 and for 9.

If the number you create ends with a zero, it will also be divisible by 5 and 10. That’s the divisibility rule for 5 and the divisibility rule for 10. AND it will also be divisible by 2 and by 6 because those are the divisibility rules for even numbers and for even numbers divisible by 3.

That leaves only three divisors to worry about: 4, 7, and 8. This post will talk about an easy and fun way to deal with those divisibility rules!

Did you know that no matter how many digits a number has, if the last digit is 2 or 6 and is preceded by an odd number, then that number will be divisible by 4? Also if the last digit of that number is 0, 4, or 8 and is preceded by an even number, then that number will be divisible by 4 as well? That’s a divisibility rule for 4.

If that number happens to have nine digits and we multiply it by 10, then the new product will also be divisible by 8. That’s because of the divisibility rule for 4 and the number getting multiplied by 2 × 5.

So 7 is the only divisor that is making any real trouble for us! Here’s how we will deal with division by 7: We will separate our number into smaller parts, all of which are divisible by 7. It would be easiest to separate it into three 3-digit numbers and then have those 9-digits be followed by a zero.

3-Digit Multiples of 7:

Now in the chart below we have all of the 3-digit multiples of 7. I have colored in all the numbers with a zero, and all numbers that repeat any of their own digits. We won’t be using any of the numbers that have been shaded in. However, we want to use at least one of the numbers printed in green. In fact, a green number should be your first choice because it is divisible by 4, while your second and third choices will be numbers that have not been shaded in.

And just like that, you can take ANY two numbers from the chart followed by a green number and zero, and you will have a number that is divisible by all the numbers from 1 to 10.

If you take care that none of the digits repeat, which is very likely with so many choices, it will also be a number that has each of the ten digits exactly one time!

For example, if I choose 812 as my green number, and 357 as my second number, my leftover digits are 9-4-6. I can check all permutations of 9-4-6 to see if any of them are on the chart, and I notice that 469 is there so I form the number 3574698120. Here’s proof that it is divisible by every number from 1 to 10:

Now you pick three of your own 3-digit numbers, attach a zero, and see how it does!

Let’s Explore the 11-and 12-Divisibility Rules:

My chosen number is also divisible by 12 because it satisfies the 3- and the 4- divisibility rules. Is it divisible by 11?

3574698120 is NOT divisible by 11. I know that because the sum of the red numbers, 26, minus the sum of the blue numbers, 19, is 7 which is not a multiple of 11.

However, if I switch the 357 with the 469, my number still satisfies all the previous divisibility rules, but now 4693578120 has an additional factor! The sum of its red numbers, 28, minus the sum of its blue numbers, 17, is 11, clearly a multiple of 11.

The secret to having the bonus that your 10-digit number is divisible by 11, is having every other number add up to either 17 or 28. Those are the sums we should look for because 28 + 17 = 45, the total of all the digits, and 28 – 17 = 11, the difference satisfying the 11 divisibility rule.

Thus, 4693578120 is divisible by all the numbers from 1 to 12. Here’s a table showing all those divisions:

Now you give it a try! Can you also use the 3-digit multiples of 7 Chart to find your own 10-digit number that is divisible by all the numbers from 1 to 12? Try it! I think you will have fun solving this puzzle, even if you have to try several different sets of 3-digit numbers to find one that works.

Factors of 1624:

  • 1624 is a composite number.
  • Prime factorization: 1624 = 2 × 2 × 2 × 7 × 29, which can be written 1624 = 2³ × 7 × 29
  • 1624 has at least one exponent greater than 1 in its prime factorization so √1624 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1624 = (√4)(√406) = 2√406.
  • The exponents in the prime factorization are 3,1 and 1. Adding one to each exponent and multiplying we get (3 + 1)(1 + 1) (1 + 1) = 4 × 2 × 2 = 16. Therefore 1624 has exactly 16 factors.
  • The factors of 1624 are outlined with their factor pair partners in the graphic below.

Factor Tree for 1624:

There are several possible factor trees for 1624. Here’s one of them.
That one used the factor pair from 28 × 58.

Wait a Minute!

1624 has all different digits and is divisible by 4 and by 7… Does that mean we can use it to form a number with 10 different digits that will be divisible by all the numbers from 1 to 10??!!!

Yes, we can. In fact, in this case, I was able to determine the needed multiples of 7 in my head because they were all 1- or 2-digit multiples!

Here’s what I did: 1624, is divisible by both 4 and 7, so 16240 is divisible by 4, 7, and 8. The missing digits are 3, 5, 7, 8, and 9. What multiples of 7 use just those digits? How about 7, 35, and 98?! Easy Peasy!

Those ten digits can be arranged in six different ways all of which are divisible by all the numbers from 1 to 10:
7359816240,
7983516240,
3579816240,
3598716240,
9873516240, and
9835716240.

Are any of those permutations of 1234567890 also divisible by 11? Yes!!! I used pencil and paper to make sure every other digit of my chosen10-digit number added up to 28 or 17. Here are all those divisions:

It is also divisible by 13, 14, 15, and 16, but I didn’t use a divisibility rule for 13, so I didn’t include any of those divisors in the chart.

4-Digit Multiples of 28:

Here is a chart showing all the 4-digit multiples of 28. Numbers containing zero or a repeated digit have been shaded. Use the chart to pick a 4-digit number that can be placed before the zero in your 10-digit number. Then find 1-, 2-, or 3-digit multiples of 7 to make your own 10-digit number that uses all the digits and is divisible by all the numbers from 1 to 10. You might even be able to satisfy the 11-divisibility rule so that your number is divisible by every number from 1 to 12. Make sure you are having fun working on this problem!

After solving the puzzle in several different ways, you might want to explore this puzzle that Andy Parkinson posted on Twitter:

Since 4967 is a relatively large prime number, 480 Factors likely can be beaten. Maybe YOU will be the one who finds a 10-different-10-digit number with more factors!

More about the Number 1624:

1624 is the hypotenuse of a Pythagorean triple:
1120-1176-1624, which is 56 times (20-21-29).

2(28)(29) = 1624. That makes 1624 four times the 28th triangular number.

1623 Easter Basket Challenge Puzzle

Today’s Puzzle:

Since I’ve recently made puzzles with a pink, purple, or blue Easter egg as well as some blades of grass blowing in the spring wind, it only makes sense that I would also give you an Easter basket in which to hold those other puzzles.

The puzzle is solved if you have written the numbers 1 to 10 in each of the boldly outlined areas of the puzzle, and if those numbers work with the clues to form four multiplication tables.

Print the puzzles or type the solution in this excel file: 12 Factors 1614-1623.

If you need a little help, here’s the same puzzle with the factor pairs for the clues written in.

And if you want even more help, here’s a 2 1/2 minute video on how to get started. I assume you already know the directions on how to solve this kind of puzzle that I gave at the top of this post.

Factors of 1623:

  • 1623 is a composite number.
  • Prime factorization: 1623 = 3 × 541.
  • 1623 has no exponents greater than 1 in its prime factorization, so √1623 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 1623 has exactly 4 factors.
  • The factors of 1623 are outlined with their factor pair partners in the graphic below.

More About the Number 1623:

1623 is the hypotenuse of a Pythagorean triple:
1023-1260-1623, which is 3 times (341-420-541).

1622 A Blue Egg for Your Easter Basket

Today’s Puzzle:

These somewhat tricky level-5 puzzles are probably better suited for middle school and up than younger kids. Use logic on every step and you should be able to find its unique solution.

Math Eggs from Twitter:

Here are some Easter egg puzzles I saw on Twitter. Some are perfect for the littles and others are for older kids. Easter egg hunts can be fun for anyone of any age.

Factors of 1622:

  • 1622 is a composite number.
  • Prime factorization: 1622 = 2 × 811.
  • 1622 has no exponents greater than 1 in its prime factorization, so √1622 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 1622 has exactly 4 factors.
  • The factors of 1622 are outlined with their factor pair partners in the graphic below.

More about the Number 1622:

1622 is the sum of four consecutive numbers:
409 + 410 + 411 + 412 = 1622.

1621 Give This Purple Egg a Crack!

Today’s Puzzle:

Here’s a level-5 purple Easter egg for you to try. All you need to do is write the numbers from 1 to 12 in both the first column and in the top row so that those numbers and the given clues function like a multiplication table. Go ahead. Give it a crack!

Factors of 1621:

  • 1621 is a prime number.
  • Prime factorization: 1621 is prime.
  • 1621 has no exponents greater than 1 in its prime factorization, so √1621 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1621 has exactly 2 factors.
  • The factors of 1621 are outlined with their factor pair partners in the graphic below.

How do we know that 1621 is a prime number? If 1621 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1621. Since 1621 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1621 is a prime number.

More about the Number 1621:

OEIS.org informs us that 1621 is in an interesting group of prime numbers.

I have verified it. They really are all prime!

1621 is the sum of two squares:
39² + 10² = 1621.

1621 is the hypotenuse of a Pythagorean triple:
780-1421-1621, calculated from 2(39)(10), 39² – 10², 39² + 10².

Here’s another way we know that 1621 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 39² + 10² = 1621 with 39 and 10 having no common prime factors, 1621 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1621. Since 1621 is not divisible by 5, 13, 17, 29, or 37, we know that 1621 is a prime number.

Math Happens When Factors of 1620 Make Sum-Difference

Math Happens Puzzle:

Math Happens put another one of my puzzles in the Austin Chronicle and the Orange Leader!

You can see the puzzle on page 23 of this e-edition or this pdf of the newspaper. You can also find links to all of my Sum-Difference puzzles here.

Here is another one of Math Happen’s amazing puzzles. The tabletops shown are exactly the same. Click on it to see proof!

And how about this way:

Math Happens in many different ways as you can see in their blog post from February 5. You can also look for Math Happens on a page in the middle of each of these  2020 issues or 2021 issues of the Austin Chronicle newspaper online.

Would you like puzzles like these in your community newspaper? Have your paper contact Math Happens on Twitter and make it happen!

Today’s Sum-Difference Puzzles:

Just like the number 6 in the newspaper puzzle above, 180 and 1620 both have factor pairs that make sum-difference. To help you solve these puzzles, I’ve listed all of their factor pairs in the graphics below the puzzle.

As shown below, 180 has nine factor pairs. One of those pairs adds up to 41, and another one subtracts to 41. Put the factors in the appropriate boxes in the first puzzle.

The needed factors for the second puzzle are multiples of the numbers in the first puzzle. 1620 has fifteen factor pairs. One of the factor pairs adds up to ­123, and a different one subtracts to 123. If you can identify those factor pairs, then you can solve the second puzzle!

What Are the Factors of 1620?

  • 1620 is a composite number.
  • Prime factorization: 1620 = 2 × 2 × 3 × 3 × 3 × 3 × 5, which can be written 1620 = 2² × 3⁴ × 5.
  • 1620 has at least one exponent greater than 1 in its prime factorization so √1620 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1620 = (√324)(√5) = 18√5.
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each exponent and multiplying we get (2 + 1)(4 + 1)(1 + 1) = 3 × 5 × 2 = 30. Therefore 1620 has exactly 30 factors.
  • The factors of 1620 are outlined with their factor pair partners in the graphic above.

Factor Trees for 1620:

Here are two of the MANY possible factor trees for 1620:

Hint: I chose to build these factor trees with those two sets of factor pairs for a reason.

More about the Number 1620:

1620 is the sum of the interior angles of a hendecagon (11-sided polygon) because
(11 – 2)180 = 1620.

1620 is the sum of two squares:
36² + 18² = 1620.

1620 is the hypotenuse of a Pythagorean triple:
972-1296-1620, calculated from 36² – 18², 2(36)(18), 36² + 18².
It is also (3-4-5) times 324.

This is only some of the math that happens with the number 1620.

1619 A Pink Egg Hidden in the Grass

Today’s Puzzle:

Easter is less than two weeks away. This pink puzzle is the first of three level-5 Easter eggs hidden amongst some blades of grass for you to find and solve. The puzzle might be a little tricky, but use logic every step of the way, and you’ll be able to find the unique solution:

Factors of 1619:

  • 1619 is a prime number.
  • Prime factorization: 1619 is prime.
  • 1619 has no exponents greater than 1 in its prime factorization, so √1619 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1619 has exactly 2 factors.
  • The factors of 1619 are outlined with their factor pair partners in the graphic below.

How do we know that 1619 is a prime number? If 1619 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1619. Since 1619 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1619 is a prime number.

More about the Number 1619:

1619 is the sum of two consecutive numbers:
809 + 810 = 1619.

1619 is also the difference of two consecutive squares:
810² – 809² = 1619.

What do you think about that?

1618 Math Happens in the Austin Chronicle

Math Happens!

Several years before I started blogging, I tried to get my puzzles in newspapers, but the publishers of those newspapers just ignored them. Because of that, it is even sweeter to me that Math Happens put one of them in the Austin Chronicle! You can see it in the newspaper on page 25 of this pdf or in this cool page-turning e-edition. Math Happens in many different ways as you can see in their blog post from February 5. You can also look for Math Happens on a page in the middle of each of these  2020 issues or 2021 issues of the Austin Chronicle newspaper online.

Math Happens also in the Orange Leader, and they would love to also be in your local community newspaper.

You can have your local newspaper contact them through Twitter!

Today’s Puzzle:

Spring happens in just a few days! Today’s puzzle represents grasses blowing in a spring wind, readily anticipating the hiding of Easter eggs. It’s a level 3 puzzle, so start by finding the factors of the clue at the top of the puzzle (and the clue that goes with it), and work your way down cell by cell until you have written all the numbers from 1 to 12 in both the factor column and the factor row. You can do this!

Factors of 1618:

  • 1618 is a composite number.
  • Prime factorization: 1618 = 2 × 809.
  • 1618 has no exponents greater than 1 in its prime factorization, so √1618 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 1618 has exactly 4 factors.
  • The factors of 1618 are outlined with their factor pair partners in the graphic below.

More about the Number 1618:

1618 = 2 × 809, and 2809 is a perfect square. Thank you OEIS.org for that fun fact!

1618 is the sum of two squares:
33² + 23² = 1618.

1618 is the hypotenuse of a Pythagorean triple:
560-1518-1618, calculated from 2(33)(23), 33² – 23², 33² + 23².
It is also 2 times (280-759-809).

1618 is the 22nd centered heptagonal number because it is one more than seven times the 21st triangular number:
7(21)(22)/2 + 1 = 1618.

1618 has exactly four factors. The last number with exactly four factors was 1603. That’s the biggest gap so far between two numbers with exactly four factors!
(It will be interesting to see who will win the horse race for the current set of 100 numbers. So far, the horses for 2 factors and 8 factors are each running twice as fast as the horse for 4 factors, and 1619 will be a prime number, giving 2 factors the lead!)

A lot of math is happening with this number!

1617 is a Pentagonal Number

Today’s Puzzle:

1617 is the 33rd pentagonal number because it is the 33 more than three times the 32nd triangular number. You can see the formula for that in the graphic below.

Can you draw lines to form three triangles each with 32 dots on its base and have exactly 33 dots left over? There are several different ways to solve this puzzle. You can scroll down to see one of those ways.

Factors of 1617:

1 + 1 + 7 = 9, a number divisible by 3, so 1617 is divisible by 3. (It wasn’t necessary to add the 6 because we already know 6 is divisible by 3.)
1 – 6 + 1 – 7 = -11, so 1617 is divisible by 11.

  • 1617 is a composite number.
  • Prime factorization: 1617 = 3 × 7 × 7 × 11, which can be written 1617 = 3 × 7² ×
  • 1617 has at least one exponent greater than 1 in its prime factorization so √1617 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1617 = (√49)(√33) = 7√33.
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each exponent and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1617 has exactly 12 factors.
  • The factors of 1617 are outlined with their factor pair partners in the graphic below.

More about the Number 1617:

1617 is the difference of two squares in SIX different ways:
809² – 808² = 1617,
271² – 268² = 1617,
119² – 112² = 1617,
79² – 68² = 1617,
49² – 28² = 1617, and
41² – 8² = 1617.

One Solution to the Puzzle:

Here is one of several possible solutions to the puzzle. Click on it if you want to see it more clearly.

1616 Centering the Pendulum

Today’s Puzzle:

Centering the Pendulum is Bill Davidson’s podcast about educators and how they inspire students to learn mathematics. All of his podcasts are wonderful and more than worth the 15 minutes or so needed to listen to each one.  I am quite honored that Find the Factors is the subject of his fourth podcast, and you can listen to it here.

To mark this occasion, I’ve made a mystery level puzzle that resembles a swinging pendulum that hopefully is centered! The puzzle might or might not be a little tricky, but just use logic every step of the way, and you should be fine.

Using logic, write the numbers from 1 to 12 in both the first column and the top row so that those numbers and the given clues function like a multiplication table. Have fun!

Factors of 1616:

  • 1616 is a composite number.
  • Prime factorization: 1616 = 2 × 2 × 2 × 2 × 101, which can be written 1616 = 2⁴ × 101
  • 1616 has at least one exponent greater than 1 in its prime factorization so √1616 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1616 = (√16)(√101) = 4√100.
  • The exponents in the prime factorization are 4 and 1. Adding one to each exponent and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1616 has exactly 10 factors.
  • The factors of 1616 are outlined with their factor pair partners in the graphic below.

More about the Number 1616:

1616 is the sum of two squares:
40² + 4² = 1616.

1616 is also the hypotenuse of a Pythagorean triple:
320-1584-1616, which is 16 times (20-99-101)
and can be calculated from 2(40)(4), 40² – 4², 40² + 4².

1616 is also the difference of two squares in three ways:
405² – 403² = 1616,
204² – 200² = 1616, and
105² – 97² = 1616.