A Multiplication Based Logic Puzzle

Archive for April, 2014

104 and Level 3

104 is a composite number. 104 = 1 x 104, 2 x 52, 4 x 26, or 8 x 13. Factors of 104: 1, 2, 4, 8, 13, 26, 52, 104. Prime factorization: 104 = 2 x 2 x 2 x 13, which can also be written 2³ x 13.

104 is never a clue in the FIND THE FACTORS puzzles.

2014-17 Level 3

The week’s puzzles and the previous week’s solutions: 10 Factors 2014-04-28

A Logical Approach to find the factors: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column and factor row.  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the table one row at a time as you go:

2014-17 Level 3 factors

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103 and Level 2

  • 103 is a prime number.
  • Prime factorization: 103 is prime.
  • The exponent of prime number 103 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 103 has exactly 2 factors.
  • Factors of 103: 1, 103
  • Factor pairs: 103 = 1 x 103
  • 103 has no square factors that allow its square root to be simplified. √103 ≈ 10.14889

How do we know that 103 is a prime number? If 103 were not a prime number, then it would be divisible by at least one prime number less than or equal to √103 ≈ 10.1 Since 103 cannot be divided evenly by 2, 3, 5, or 7, we know that 103 is a prime number.

103 is never a clue in the FIND THE FACTORS puzzles.

2014-17 Level 2

This week’s puzzles and last week’s factors: 10 Factors 2014-04-28

2014-17 Level 2 factors

102 and Level 1

102 is a composite number. 102 = 1 x 102, 2 x 51, 3 x 34, or 6 x 17. Factors of 102: 1, 2, 3, 6, 17, 34, 51, 102. Prime factorization: 102 = 2 x 3 x 17.

102 is never a clue in the FIND THE FACTORS puzzles.

2014-17 Level 1

This week’s puzzles and last week’s solutions: 10 Factors 2014-04-28

2014-17 Level 1 factors

101 and Level 6

  • 101 is a prime number.
  • Prime factorization: 101 is prime.
  • The exponent of prime number 101 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 101 has exactly 2 factors.
  • Factors of 101: 1, 101
  • Factor pairs: 101 = 1 x 101
  • 101 has no square factors that allow its square root to be simplified. √101 ≈ 10.0498756

How do we know that 101 is a prime number? If 101 were not a prime number, then it would be divisible by at least one prime number less than or equal to √101 ≈ 10. Since 101 cannot be divided evenly by 2, 3, 5, or 7, we know that 101 is a prime number.

101 is never a clue in the FIND THE FACTORS puzzles.

2014-16 Level 6

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

2014-16 Level 6 Logic

The 100th Day of School

Kindergarten and first grade students from coast to coast celebrate the 100th Day of school. They read books about 100 and the 100th day of school. They bring collections of 100 items to school, and they also complete many 100th day worksheets.

My blogging friend, Paula Krieg, designed a fabulous art project for second grade students. I just had to update this post and share her work with you. Her project would also be perfect for the 100th day of school for the mid-elementary grades. In this video, she explains her 100¢ art project:

Her post includes detailed instructions, templates of everything needed to complete the project, and links showing completed projects. Check it out!

I like the following Hap Palmer song about coins. It happens to teach important multiplication facts, including some about the number 100. Third graders could learn these multiplication facts on the 100th day of school. This song would also go very well with Paula’s 100¢ project above:

Five pennies make a nickel
Two nickels make a dime
Ten dimes make a dollar
That’s a hundred pennies in all

Five pennies make a nickel
Five nickels make a quarter
Four quarters make a dollar
That’s a hundred pennies in all

What about students in 4th or 5th grade or even middle school or high school? Can older students celebrate the 100th day of school? I think so! Share some of the following information with them:

Here is the basic factoring information for the number 100.

  • 100 is a composite number.
  • Prime factorization: 100 = 2 × 2 × 5 × 5, which can be written 100 = 2²× 5²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 100 has exactly 9 factors.
  • Factors of 100: 1, 2, 4, 5, 10, 20, 25, 50, 100
  • Factor pairs: 100 = 1 x 100, 2 x 50, 4 x 25, 5 x 20, or 10 x 10
  • 100 is a perfect square. √100 = 10

Here are a few of several possible factor trees that can be made for the number 100:

What do you think the area of the following shape will be?

You got it, 100 cm².

Since √100 = 10, a whole number, 100 is a perfect square. Did you know that 100 can be written as the sum of TWO OTHER perfect squares?

  • 36 + 64 = 100. In other words, 6² + 8² = 10²

36, 64, and 100 were all included as clues in the very first puzzle I published on this blog:343-1

Note: When 100 is a clue in a FIND THE FACTORS puzzles, always use 10 × 10. Even though 100 has other factors, 10 × 10 is its only factor pair in which both factors are ten or less.

100 is the sum of a few different types of consecutive numbers:

  • 100 = 18 + 19 + 20 + 21 + 22; that’s the sum of 5 consecutive numbers
  • 100 = 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16; that’s the sum of 8 consecutive numbers
  • 100 = 22 + 24 + 26 + 28; that’s the sum of 4 consecutive EVEN numbers
  • 16 + 18 + 20 + 22 + 24 = 100; that’s the sum of 5 consecutive EVEN numbers
  • 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 + 17 + 19 = 100; that’s the sum of the first 10 ODD numbers

  • (1 + 2 + 3 + 4)² = 100; that the square of the sum of the first four numbers
  • 100 = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23; that’s the sum of the first 9 prime numbers.
  • 100 = 47 + 53; that’s the sum of two consecutive prime numbers
  • 1³ + 2³ + 3³ + 4³ = 100, that’s the sum of the first four cubes

100 is the hypotenuse of two Pythagorean triple triangles:

  • 60² + 80² = 100²
  • 28² + 96² = 100²

But 100 is a leg in SEVERAL other Pythagorean triples:

  • 75² + 100² = 125²
  • 100² + 105² = 145²
  • 100² + 240² = 260²
  • 100² + 495² = 505²
  • 100² + 621² = 629²
  • 100² + 1248² = 1252²
  • 100² + 2499² = 2501²

That last one is called a Primitive Pythagorean triple because the greatest common factor of 100, 2499, and 2501 is one. Can you find the greatest common factor of any of the other triples?

Here’s the ONE non-trivial way 100 is the difference of two squares:

  • 26² – 24² = 100, and that’s because (26 + 24)(26 – 24) = 100

100 is a palindrome in five different bases (Palindromes read the same backwards or forwards such as “Madam, I’m Adam” or “race car”):

  • 10201 in BASE 3 because 1(3⁴) + 0(3³) + 2(3²) + 0(3¹) + 1(3º) = 100
  • 202 in BASE 7 because 2(7²) + 0(7¹) + 2(7º) = 100
  • 121 BASE 9 because 1(9²) + 2(9¹) + 1(9º) = 100
  • 55 BASE 19 because 5(19¹) + 5(19º) = 100
  • 44 BASE 24 because 4(24¹) + 4(24º) = 100

If you convert 100 from base 10 to these other bases, you will get:

  • 40 in BASE 25
  • 50 in BASE 20
  • 400 in BASE 5

We can find numbers that have 100 factors by using prime factorizations. This chart lists some candidates for the smallest number with 100 factors. Notice that to make each option as small as possible, the largest exponent goes with the smallest prime number in each prime factorization. Can you order these nine numbers from smallest to greatest?

The smallest number to have exactly 100 factors is 45,360. Can you list all 100 of those factors?

Now for today’s puzzle. . .100 isn’t one of its clues, but can you find the one and only place where 100 belongs in this mixed up multiplication table?

2014-16 Level 5

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

I hope you fully enjoy the 100th day of school!

2014-16 Level 5 Logic

99 and Level 4

99 is a composite number. 99 = 1 x 99, 3 x 33, or 9 x 11. Factors of 99: 1, 3, 9, 11, 33, 99. Prime factorization: 99 = 3 x 3 x 11, which can also be written 99 = 3² x 11.

Sometimes 99 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, we use only 9 x 11 in the puzzles.

2014-16 Level 4

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

2014-16 Level 4 Logic

98 and What Do You Do When You Get to the End of a Shoe Lace?

98 is a composite number. 98 = 1 x 98, 2 x 49, or 7 x 14. Factors of 98: 1, 2, 7, 14, 49, 98. Prime factorization: 98 = 2 x 7 x 7, which can also be written 98 = 2 x 7²

98 is never a clue in the FIND THE FACTORS puzzles.

In the past finding the factors for a level 3 puzzles has been like lacing  an entire shoe with only one end of the lace. Today’s puzzle is slightly more difficult because that end of the lace gets cut off in the middle of the puzzle, and the other end of the lace has to be used to find the rest of the factors. Level 3 is meant to be a bridge between the easier levels and the higher leveled puzzles, and those laces get cut off in the higher leveled puzzles all the time. Good luck with this level 3 puzzle! I’m sure you can still find all the factors!

2014-16 Level 3

This week’s puzzles and last week’s solutions: 12 Factors 2014-04-21

A Logical Approach to find the factors: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column and factor row.  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the table one row at a time as you go:

2014-16 Level 3 Factors

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