1633 and Level 5

Today’s Puzzle:

It might be tricky in a few places, but use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues behave like a multiplication table.

Factors of 1633:

  • 1633 is a composite number.
  • Prime factorization: 1633 = 23 × 71.
  • 1633 has no exponents greater than 1 in its prime factorization, so √1633 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 1633 has exactly 4 factors.
  • The factors of 1633 are outlined with their factor pair partners in the graphic below.

More about the Number 1633:

1633 is the difference of two squares in two different ways:
817² – 816² = 1633, and
47² – 24² = 1633.

Math Happens When Two of 1632’s Factors Look in a Mirror!

Today’s Puzzle:

Both 12 and 102 are factors of 1632. Something special happens when either one squares itself and looks in a mirror. Solving this puzzle from Math Happens will show you what happens to 12 and 12².

You can see that puzzle on page 33 of this e-edition or this pdf of the Austin Chronicle. You can find other Math Happens Puzzles here.

This next puzzle will help you discover what happens when 102 and 102² look in a mirror!

Why do you suppose the squares of (12, 21) and (102, 201) have that mirror-like property?

Factor Trees for 1632:

There are many possible factor trees for 1632, but today I will focus on two trees that use factor pairs containing either 12 or 102:

Factors of 1632:

  • 1632 is a composite number.
  • Prime factorization: 1632 = 2 × 2 × 2 × 2 × 2 × 3 × 17, which can be written 1632 = 2⁵ × 3 × 17.
  • 1632 has at least one exponent greater than 1 in its prime factorization so √1632 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1632 = (√16)(√102) = 4√102.
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each exponent and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1632 has exactly 24 factors.
  • The factors of 1632 are outlined with their factor pair partners in the graphic below.

More about the Number 1632:

1632 is the hypotenuse of a Pythagorean triple:
768-1440-1632, which is (8-15-17) times 96.

1632 is the difference of two squares in EIGHT different ways:
409² – 407² = 1632,
206² – 202² = 1632,
139² – 133² = 1632,
106² – 98² = 1632,
74² – 62² = 1632,
59² – 43² = 1632,
46² – 22² = 1632, and
41² – 7² = 1632.

That last difference of two squares means 1632 is only 49 numbers away from the next perfect square, 1681.

 

1631 The Importance of Practice

Today’s Puzzle:

I did not have the privilege of learning a musical instrument when I was growing up, but I did make sure my children had that opportunity. One of the topics discussed in this next episode of Bill Davidson’s Podcast is the importance that practice plays in both music and mathematics. I thought it was quite good.

I think practicing is best when it is enjoyable. If you solve this musical note puzzle, it will hopefully be an enjoyable way for you to practice a few multiplication and division facts. Just use logic to write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues will function like a multiplication table.

Factors of 1631:

  • 1631 is a composite number.
  • Prime factorization: 1631 = 7 × 233.
  • 1631 has no exponents greater than 1 in its prime factorization, so √1631 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 1631 has exactly 4 factors.
  • The factors of 1631 are outlined with their factor pair partners in the graphic below.

More about the Number 1631:

1631 is the hypotenuse of a Pythagorean triple:
735-1456-1631, which is 7 times (105-208-233).

1631 is the difference of two squares in two different ways:
816² – 815² = 1631, and
120² – 113² = 1631.

I found those number facts just from looking at the factors of 1631.

 

1630 and Level 3

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so those numbers and the given clues make the puzzle function like a multiplication table. Because this is a level 3 puzzle, first write the factors for 72 and 90. Then work your way down the puzzle row by row until you have found all the factors.

Factors of 1630:

  • 1630 is a composite number.
  • Prime factorization: 1630 = 2 × 5 × 163.
  • 1630 has no exponents greater than 1 in its prime factorization, so √1630 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 1630 has exactly 8 factors.
  • The factors of 1630 are outlined with their factor pair partners in the graphic below.

More about the Number 1630:

1630 is the hypotenuse of a Pythagorean triple:
978-1304-1630, which is (3-4-5) times 326.

1629 and Level 2

Today’s Puzzle:

Write the numbers from 1 to 10 in both the first column and the top row so that those numbers and the given clues work to make a multiplication table.

Factors of 1629:

1 + 6 + 2 = 9, so 1929 is divisible by both 3 and 9. (It’s only necessary to add the non-nine numbers together to check those two divisibility rules.)

  • 1629 is a composite number.
  • Prime factorization: 1629 = 3 × 3 × 181, which can be written 1629 = 3² × 181.
  • 1629 has at least one exponent greater than 1 in its prime factorization so √1629 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1629 = (√9)(√181) = 3√181.
  • 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 1629 has exactly 6 factors.
  • The factors of 1629 are outlined with their factor pair partners in the graphic below.

More about the Number 1629:

1629 is the sum of two squares:
30² + 27² = 1629.

1629 is the hypotenuse of a Pythagorean triple:
171-1620-1629, calculated from 30² – 27², 2(30)(27), 30² + 27².
It is also 9 times (19-180-181).

1628 A Simple Cross

Today’s Puzzle:

A simple cross is an appropriate symbol for Good Friday. Write the numbers from 1 to 10 in the first column and the top row so that those numbers and the given clues function like a multiplication table.

Factors of 1628:

  • 1628 is a composite number.
  • Prime factorization: 1628 = 2 × 2 × 11 × 37, which can be written 1628 = 2² × 11 × 37.
  • 1628 has at least one exponent greater than 1 in its prime factorization so √1628 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1628 = (√4)(√407) = 2√407.
  • 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 1628 has exactly 12 factors.
  • The factors of 1628 are outlined with their factor pair partners in the graphic below.

More about the Number 1628:

1628 is the hypotenuse of a Pythagorean triple:
528-1540-1628, which is (12-35-37) times 44.

1628 is the difference of two squares in two different ways:
408² – 406² = 1628, and
48² – 26²  = 1628.

1627 Color-Coded Prime Numbers

Today’s Puzzle:

Study this color-coded chart of prime numbers. 1627 is the smallest prime number that begins something special. Can you figure out what that is?

Also, why do you think I’ve underlined some of the other prime numbers on the list?

Memorizing which numbers are prime can be a big time-saver in mathematics. How many in a row can you recite without looking?

Factors of 1627:

  • 1627 is a prime number.
  • Prime factorization: 1627 is prime.
  • 1627 has no exponents greater than 1 in its prime factorization, so √1627 cannot be simplified.
  • The exponent in the prime factorization is 1. Adding one to that exponent we get (1 + 1) = 2. Therefore 1627 has exactly 2 factors.
  • The factors of 1627 are outlined with their factor pair partners in the graphic below.

How do we know that 1627 is a prime number? If 1627 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1627. Since 1627 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, or 37, we know that 1627 is a prime number.

More about the Number 1627:

1627 is the sum of two consecutive numbers:
813 + 814 = 1627.

1627 is also the difference of two consecutive squares:
814² – 813² = 1627.

What do you think of that?

1626 is a Centered Pentagonal Number

Today’s Puzzle:

1626 is the 26th centered pentagonal number and it is also one more than 5 times the 25th triangular number.

Today’s puzzle is for you to figure out why the following two expressions are equivalent. The first expression is the formula for the nth centered pentagonal number, and the second expression is one more than 5 times the formula for the nth triangular number.
(5n²-5n+2)/2, when n = 26, and
1 + 5n(n+1)/2, when n = 25.

Factors of 1626:

  • 1626 is a composite number.
  • Prime factorization: 1626 = 2 × 3 × 271.
  • 1626 has no exponents greater than 1 in its prime factorization, so √1626 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 1626 has exactly 8 factors.
  • The factors of 1626 are outlined with their factor pair partners in the graphic below.

More about the Number 1626:

2(813)(1) = 1626 means that 2(813)(1), 813² – 1², 813² + 1² is a Pythagorean triple, and
2(271)(3) = 1626 means that 2(271)(3), 271² – 3², 271² + 3² is as well.

Calculate those expressions and you will have found the only two Pythagorean triples containing the number 1626.

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.