# 1408 Powers of 2 in the Multiplication Table

number, puzzle, factors, factor pairs, prime factorization,

I have a 10 × 10 multiplication table poster in my classroom to help students who haven’t memorized the times’ table yet. We have to spend our time going over more advanced topics. One student struggled with the idea of raising two to a power. I went to the poster and boxed in all the powers of two on it. While I boxed them in, I recited, “2⁰ = 1, 2¹ = 2, 2² = 2×2 = 4, 2³ = 2×2×2 = 8, 2⁴ = 2×2×2×2= 16, 2⁵ = 2×2×2×2×2= 32, 2⁶ = 2×2×2×2×2×2=64.”

I liked the pattern those powers of two made on the poster so I made this 32×32 multiplication chart on my computer and continued the pattern. I expect the chart has many things for you to notice and wonder about. You could also do it with powers of 3, or another number, but you would need to use a much bigger multiplication table to show as many powers.

Now I’ll tell you a little bit about the number 1408.

1408 is not a power of 2, but it is 11 times a power of 2, specifically, it is 11 × 2⁷.

• 1408 is a composite number.
• Prime factorization: 1408 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 11, which can be written 1408 = 2⁷ × 11
• 1408 has at least one exponent greater than 1 in its prime factorization so √1408 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1408 = (√64)(√22) = 8√22
• The exponents in the prime factorization are 7 and 1. Adding one to each exponent and multiplying we get (7 + 1)(1 + 1) = 8 × 2 = 16. Therefore 1408 has exactly 16 factors.
• The factors of 1408 are outlined with their factor pair partners in the graphic below. Here is a festive multilayered factor cake for 1408: So delicious! And here is a nicely balanced factor tree showing all of its prime factors: # 1350 Logic is at the Heart of This Puzzle

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

If you need some tips on how to get started on this puzzle, check out this video:

Now I’ll tell you a few things about the number 1350:

• 1350 is a composite number.
• Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
• The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
• Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
• Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
• Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

# 1226 Happy Birthday to My Sister, Sue

I don’t make puzzles bigger than 12 × 12 very often, but I decided to make this one, a 17 × 17 Mystery Level for my sister’s birthday. I know she can solve smaller ones without any problems, so I wanted to give her a challenge. Happy birthday, Sue. I hope you have a great day and enjoy solving this one. Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Note that with a bigger table there are several more possible common factors:

Is 4, 8, or 16 the common factor needed for 64 and 32 or for 16 and 48?
Is 7 or 14 the common factor needed for 14 and 70?
Is 6, 10, or 15 the common factor needed for 60 and 90?

As always there is only one solution. The table below will help anyone not familiar with some of the lesser known multiplication facts needed to solve the puzzle. Now I’ll share some information about the number 1226:

• 1226 is a composite number.
• Prime factorization: 1226 = 2 × 613
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1226 has exactly 4 factors.
• Factors of 1226: 1, 2, 613, 1226
• Factor pairs: 1226 = 1 × 1226 or 2 × 613
• 1226 has no square factors that allow its square root to be simplified. √1226 ≈ 35.01428 35² + 1² = 1226

1226 is the hypotenuse of a Pythagorean triple:
70-1224-1226 calculated from 2(35)(1), 35² – 1², 35² + 1²

# 1161 and Level 1

Solving this puzzle will help you review the multiplication table. Knowing the multiplication table inside and out will be a big PLUS in your life. It will save you so much time in all your mathematics classes! Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Here is some information about the number 1161:

• 1161 is a composite number.
• Prime factorization: 1161 = 3 × 3 × 3 × 43, which can be written 1161 = 3³ × 43
• The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1161 has exactly 8 factors.
• Factors of 1161: 1, 3, 9, 27, 43, 129, 387, 1161
• Factor pairs: 1161 = 1 × 1161, 3 × 387, 9 × 129, or 27 × 43
• Taking the factor pair with the largest square number factor, we get √1161 = (√9)(√129) = 3√129 ≈ 34.07345 1161 is the sum of the first twenty-six prime numbers. That’s all the primes from 2 to 101.

1161 is a palindrome in a couple of bases:
It’s 10010001001 in BASE 2 because 2¹⁰ + 2⁷ + 2³ + 2⁰ = 1161 and
1B1 in BASE 29 (B is 11 base 10) because 29² + 11(29) + 1 = 1161

# 1097 and Level 3

72 and 27 are mirror images of each other. What is the largest number that will divide evenly into both of them? Put the answer to that question under the x, and you will have completed the first step in solving this multiplication table puzzle. Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Here’s a little bit more about the number 1097:

• 1097 is a prime number.
• Prime factorization: 1097 is prime.
• The exponent of prime number 1097 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1097 has exactly 2 factors.
• Factors of 1097: 1, 1097
• Factor pairs: 1097 = 1 × 1097
• 1097 has no square factors that allow its square root to be simplified. √1097 ≈ 33.12099

How do we know that 1097 is a prime number? If 1097 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1097 ≈ 33.1. Since 1097 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1097 is a prime number. 1097 is the final prime number in the prime triplet, 1091-1093-1097.

1097 is the sum of two squares:
29² + 16² = 1097

1097 is the hypotenuse of a primitive Pythagorean triple:
585-928-1097 calculated from 29² – 16², 2(29)(16), 29² + 16²

Here’s another way we know that 1097 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 16² = 1097 with 29 and 16 having no common prime factors, 1097 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1097 ≈ 33.1. Since 1097 is not divisible by 5, 13, 17, or 29, we know that 1097 is a prime number.

# 1087 and Level 3

Using logic, start with the clue on the top row and work yourself down row by row filling in the appropriate factors while you go. You might find this level 3 puzzle a little tricky near the bottom of the puzzle, so I didn’t want to wait to share it with you. Happy factoring! Print the puzzles or type the solution in this excel file: 10-factors-1087-1094

1087 is the first prime since 1069, which was 18 numbers ago! What else can I tell you about it?

• 1087 is a prime number.
• Prime factorization: 1087 is prime.
• The exponent of prime number 1087 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1087 has exactly 2 factors.
• Factors of 1087: 1, 1087
• Factor pairs: 1087 = 1 × 1087
• 1087 has no square factors that allow its square root to be simplified. √1087 ≈ 32.96968

How do we know that 1087 is a prime number? If 1087 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1087 ≈ 33. Since 1087 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1087 is a prime number. 1087 is also palindrome 767 in BASE 12 because
7(12²) + 6(12) + 7(1) = 1087

# 1057 and Level 2

Can you figure out where to put the numbers from 1 to 10 in both the 1st column and the top row so that this puzzle behaves like a multiplication table? Print the puzzles or type the solution in this excel file: 10-factors-1054-1062

Here’s a little bit about the number 1057:

• 1057 is a composite number.
• Prime factorization: 1057 = 7 × 151
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1057 has exactly 4 factors.
• Factors of 1057: 1, 7, 151, 1057
• Factor pairs: 1057 = 1 × 1057 or 7 × 151
• 1057 has no square factors that allow its square root to be simplified. √1057 ≈ 32.5115 # 1049 and Level 6

Find the Factors Puzzles are always solved using logic. Can you see the logic needed to solve this one? Print the puzzles or type the solution in this excel file: 12 factors 1044-1053

Here are a few facts about the number 1049:

• 1049 is a prime number.
• Prime factorization: 1049 is prime.
• The exponent of prime number 1049 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1049 has exactly 2 factors.
• Factors of 1049: 1, 1049
• Factor pairs: 1049 = 1 × 1049
• 1049 has no square factors that allow its square root to be simplified. √1049 ≈ 32.38827

How do we know that 1049 is a prime number? If 1049 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1049 ≈ 32.4. Since 1049 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1049 is a prime number. 1049 is also the sum of three consecutive prime numbers:
347 + 349 + 353 = 1049

32² + 5² = 1049 so 1049 is the hypotenuse of a Pythagorean triple:
320-999-1049 calculated from 2(32)(5), 32² – 5², 32² + 5²

Here’s another way we know that 1049 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 5² = 1049 with 32 and 5 having no common prime factors, 1049 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1049 ≈ 32.4. Since 1049 is not divisible by 5, 13, 17, or 29, we know that 1049 is a prime number.

# 1047 and Level 4

There are a couple of clues in this puzzle that might be a little tricky, but I know you won’t let that stop you from finding its solution. Puzzles are fun, so have fun with this one. Print the puzzles or type the solution in this excel file: 12 factors 1044-1053

What do I know about the number 1047?

• 1047 is a composite number.
• Prime factorization: 1047 = 3 × 349
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1047 has exactly 4 factors.
• Factors of 1047: 1, 3, 349, 1047
• Factor pairs: 1047 = 1 × 1047 or 3 × 349
• 1047 has no square factors that allow its square root to be simplified. √1047 ≈ 32.357379 1047 is the hypotenuse of a Pythagorean triple:
540-897-1047 which is 3 times (180-299-349)

It is also a palindrome in a couple of bases:
It’s 343 in BASE 18 because 3(18²) + 4(18) + 3(1) = 1047, and
2H2 in BASE 19 (H is 17 base 10) because 2(19²) + 17(19) + 2(1) = 1047

# 1046 and Level 3

To solve this Level 3 puzzle start with the clues in the first row, 15 and 5. Put their factors in the first column and top row, then work down the puzzle finding the factors of all of the clues. Every factor you write in the first column or top row must be a number from 1 to 12 and can only be used once in each place. Now here is a little bit about the number 1046:

• 1046 is a composite number.
• Prime factorization: 1046 = 2 × 523
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1046 has exactly 4 factors.
• Factors of 1046: 1, 2, 523, 1046
• Factor pairs: 1046 = 1 × 1046 or 2 × 523
• 1046 has no square factors that allow its square root to be simplified. √1046 ≈ 32.34192 1046 is also palindrome 626 in BASE 13 because 6(13²) + 2(13) + 6 (1) = 1046