## A Multiplication Based Logic Puzzle

### 700 Pick Your Pony! Who will win this Amount of Factors Horse Race?

• 700 is a composite number.
• Prime factorization: 700 = 2 x 2 x 5 x 5 x 7, which can be written 700 = (2^2) x (5^2) x 7
• The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 700 has exactly 18 factors.
• Factors of 700: 1, 2, 4, 5, 7, 10, 14, 20, 25, 28, 35, 50, 70, 100, 140, 175, 350, 700
• Factor pairs: 700 = 1 x 700, 2 x 350, 4 x 175, 5 x 140, 7 x 100, 10 x 70, 14 x 50, 20 x 35 or 25 x 28
• Taking the factor pair with the largest square number factor, we get √700 = (√100)(√7) = 10√7 ≈ 26.457513.

Because this is my 700th post, I think I’ll have another horse race. Some numbers from 601 to 700 have exactly 2 factors, 4 factors, and so forth up to 24 factors. (Only perfect squares can have an odd number of factors.)

Which number from 1 to 24 will win this amount of factors horse race? Which number will come in second place, or third place? Cheering for more than one pony will make the race even more interesting.

Here we see that the numbers 2, 6, & 8 are the first ones out of the gate. Click on the graphic to see the rest of this very thrilling horse race:

Every hundred posts I also like to focus on the percentage of numbers whose square roots can be simplified.

700 is divisible by 100 so its square root can easily be simplified: √700 = 10√7.

273 of the first 700 numbers have reducible square roots. That’s exactly 39%.

The rest of the numbers, 427, which is 41% of the first 700 numbers, do not have reducible square roots.

Here’s a table breaking down the amount of factors in each group of one hundred integers and the number of reducible square roots.

Here are some facts about the number 700.

700 is a palindrome in several bases:

• 4A4 BASE 12; note A is equivalent to 1o in base 10, and 4(144) + 10(12) + 4(1) = 700
• PP BASE 27; note P is equivalent to 25 in base 10, and 25(27) + 25(1) = 700
• KK Base 34; note K is equivalent to 20 in base 10, and 20(34) + 20(1) = 700

700 is the sum of four consecutive prime numbers: 167 + 173 + 179 + 181.

Here is a beautiful painting of a horse race that I saw on twitter:

### 508 A Mathematical and Biblical Truth: 2 > 1

Two are better than one; because they have a good reward for their labour.

For if they fall, the one will lift up his fellow: but woe to him that is alone when he falleth; for he hath not another to help him up.

Ecclesiastes 4: 9 – 10

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508 is the sum of some consecutive prime numbers, and at least one of those prime numbers is in its prime factorization. Can you figure out what those consecutive primes are?

• 508 is a composite number.
• Prime factorization: 508 = 2 x 2 x 127, which can be written 508 = (2^2) x 127
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 508 has exactly 6 factors.
• Factors of 508: 1, 2, 4, 127, 254, 508
• Factor pairs: 508 = 1 x 508, 2 x 254, or 4 x 127
• Taking the factor pair with the largest square number factor, we get √508 = (√4)(√127) = 2√127 ≈ 22.53 88 55 33 9

### 336 and Level 2

• 336 is a composite number.
• Prime factorization: 336 = 2 x 2 x 2 x 2 x 3 x 7, which can be written 336 = (2^4) x 3 x 7
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 336 has exactly 20 factors.
• Factors of 336: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336
• Factor pairs: 336 = 1 x 336, 2 x 168, 3 x 112, 4 x 84, 6 x 56, 7 x 48, 8 x 42, 12 x 28, 14 x 24 or 16 x 21
• Taking the factor pair with the largest square number factor, we get √336 = (√16)(√21) = 4√21 ≈ 18.330

These sixteen clues are all you need to complete this multiplication table!

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

### 335 and Level 1

• 335 is a composite number.
• Prime factorization: 335 = 5 x 67
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 335 has exactly 4 factors.
• Factors of 335: 1, 5, 67, 335
• Factor pairs: 335 = 1 x 335 or 5 x 67
• 335 has no square factors that allow its square root to be simplified. √335 ≈ 18.303

If you can multiply, divide and use a little logic, you should easily be able to complete this multiplication table puzzle.

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

### 334 and Level 6

• 334 is a composite number.
• Prime factorization: 334 = 2 x 167
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 334 has exactly 4 factors.
• Factors of 334: 1, 2, 167, 334
• Factor pairs: 334 = 1 x 334 or 2 x 167
• 334 has no square factors that allow its square root to be simplified. √334 ≈ 18.276

Is 1 or 2 the common factor for 6 and 8? Is 3 or 6 the common factor for 12 and 30? Is 4 or 8 the common factor for 40 and 16? In each case only one of those choices will make this puzzle work as a multiplication table? Can you figure out what those choices should be? Use logic to find the correct solution, not trial and error.

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

### 333 and Level 5

• 333 is a composite number.
• Prime factorization: 333 = 3 x 3 x 37, which can be written 333 = $3^2$ x 37
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 333 has exactly 6 factors.
• Factors of 333: 1, 3, 9, 37, 111, 333
• Factor pairs: 333 = 1 x 333, 3 x 111, or 9 x 37
• Taking the factor pair with the largest square number factor, we get √333 = (√9)(√37) = 3√37 ≈ 18.248

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

### 332 and a Christmas Star

• 332 is a composite number.
• Prime factorization: 332 = 2 x 2 x 83 which can be written $2^2$ x 83
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 332 has exactly 6 factors.
• Factors of 332: 1, 2, 4, 83, 166, 332
• Factor pairs: 332 = 1 x 332, 2 x 166, or 4 x 83
• Taking the factor pair with the largest square number factor, we get √332 = (√4)(√83) = 2√83 ≈ 18.221

Merry Christmas! This is a rather easy level 5 puzzle so I’m sharing it instead of a level 4 puzzle today.

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

Do You Hear What I Hear? by Noël Regney

Said the night wind to the little lamb,
Do you see what I see
Way up in the sky, little lamb,
Do you see what I see
A star, a star, dancing in the night
With a tail as big as a kite
With a tail as big as a kite

Said the little lamb to the shepherd boy,
Do you hear what I hear
Ringing through the sky, shepherd boy,
Do you hear what I hear
A song, a song, high above the trees
With a voice as big as the sea
With a voice as big as the sea

Said the shepherd boy to the mighty king,
Do you know what I know
In your palace warm, mighty king,
Do you know what I know
A Child, a Child shivers in the cold
Let us bring Him silver and gold
Let us bring Him silver and gold

Said the king to the people everywhere,
Listen to what I say
Pray for peace, people everywhere!
Listen to what I say
The Child, the Child, sleeping in the night
He will bring us goodness and light
He will bring us goodness and light