Today’s Puzzle:

What percentage of natural numbers less than or equal to 1784 have simplifiable square roots?

Here is a chart of the 601st to the 700th simplifiable square roots:

You can figure out the percentage of numbers up to 1784 that have simplifiable square roots by calculating 700×100 ÷1784.

Was the percentage higher or lower than you expected?

The green areas on the chart are for consecutive numbers with simplifiable square roots. 1680-1684 are the smallest five consecutive numbers that can make that claim. Why can they? Because every one of their prime factorizations has an exponent greater than one in it.

Factors of 1784:

• 1784 is a composite number.
• Prime factorization: 1784 = 2 × 2 × 2 × 167, which can be written 1784 = 2³ × 167.
• 1784 has at least one exponent greater than 1 in its prime factorization so √1784 can be simplified. Taking the factor pair from the factor pair table below with the largest square number factor, we get √1784 = (√4)(√446) = 2√446.
• 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 1784 has exactly 8 factors.
• The factors of 1784 are outlined with their factor pairs in the graphic below.

More About the Number 1784:

1784 is the difference of two squares in two different ways:
447² – 445² = 1784, and
225² – 221² = 1784.

1784 is a palindrome in two bases:
It’s 494 in base20 because 4(20²)+9(20)+4(1) = 1784, and
2C2 in base27 because 2(27²)+12(27)+2(1) = 1784.