1584 You Will L♥ve This Challenge Puzzle!

Today’s Puzzle:

Here’s a Valentine’s challenge for you: Can you write the numbers from 1 to 10 in each of the four areas of the puzzle that touch the box with the X so that the numbers you write make the puzzle become four multiplication tables? It won’t be easy, but remember to use logic. If you succeed, you will absolutely lve the puzzle!

1584 Factor Cake:

1584 obviously is divisible by 2. It is divisible by 4 because 84 is divisible by 4.
It is divisible by 8 because 84 is not divisible by 8 and 5 is odd.
1584 is divisible by 3 and by 9 because 1 + 5 + 8 + 4 = 18, a number divisible by 3 and by 9.
1584 is divisible by 11 because 1 – 5 + 8 – 4 = 0.

Thus, 1584 makes a very tall and very nice factor cake with a couple of candles on top!

Factors of 1584:

  • 1584 is a composite number.
  • Prime factorization: 1584 = 2 × 2 × 2 × 2 × 3 × 3 × 11, which can be written 1584 = 2⁴ × 3² × 5.
  • 1584 has at least one exponent greater than 1 in its prime factorization so √1584 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1584 = (√144)(√11) = 12√11.
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each exponent and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30. Therefore 1584 has exactly 30 factors.
  • The factors of 1584 are outlined with their factor pair partners in the graphic below.

More about the Number 1584:

1584 is the difference of two squares NINE different ways:
397² – 395² = 1584,
200² – 196² = 1584,
135² – 129² = 1584,
103² – 95² = 1584,
72² – 60² = 1584,
53² – 35² = 1584,
47² – 25² = 1584,
45² – 21² = 1584, and
40² – 4² = 1584.
Yes, we are only 16 numbers away from 1600, the next perfect square!

1584 is in this cool pattern:

 

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