divisibility by 11

1661 A Little About Palindromes

/ ivasallay / Leave a comment

Today’s Puzzle:

A number is a palindrome if it reads the same backward as it does forward. For example, 1661 is a 4-digit palindrome. Today’s puzzle asks you to explore number palindromes and their factors.

Half of the time when a 4-digit palindrome is divided by 11, we get a 3-digit palindrome. Why does that happen?

Are there 3-digit palindromes that were NOT included in the Divided by Eleven column in the table above?

All 2-digit palindromes are divisible by 11. They are 11, 22, 33, 44, 55, 66, 77, 88, 99.

There are only eight 3-digit palindromes that are divisible by 11. They are 121, 242, 363, 484, 616, 737, 858, 979. Some 3-digit palindromes are prime numbers. Others are divisible by 101 or 111. Still, there are plenty of 3-digit palindromes that are composite numbers but not divisible by 11, 101, or 111.

Most 5-digit palindromes are NOT divisible by 11. I was able to construct one that is, 76967, because the red digits minus the blue digits are 23 – 12 = 11, a number divisible by 11.

Here’s another: 81818. It works because 24 – 2 = 22, a number divisible by 11.

What 5-digit palindrome can you construct that is divisible by 11?

Will an N-digit palindrome be divisible by 11? What difference does it make if N is an even number or if N is an odd number?

Factors of 1661:

1661 is divisible by 11 because the red digits minus the blue digits equal 0, a number divisible by 11.

  • 1661 is a composite number.
  • Prime factorization: 1661 = 11 × 151.
  • 1661 has no exponents greater than 1 in its prime factorization, so √1661 cannot be simplified.
  • The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1661 has exactly 4 factors.
  • The factors of 1661 are outlined with their factor pair partners in the graphic below.

More about the Number 1661:

1661 is also a palindrome in base 18:
1661₁₀ = 525₁₈ because
5(18²) + 2(18¹) + 5(18º) =
5(324) + 2(18) + 5(1) =
1620 + 36 + 5 =
1661.

1067 and Level 5

/ ivasallay / Leave a comment

The common factors of 20 and 40 are 1, 2, 4, 5, and 10. Only the ones in blue will put numbers from 1 to 12 in the top row, as required. Since there is more than one possible common factor, don’t start with those two clues. This is a level 5 puzzle so at least one pair of clues will work to get you started.

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Here are a few facts about the number 1067:

  • 1067 is a composite number.
  • Prime factorization: 1067 = 11 × 97
  • 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 1067 has exactly 4 factors.
  • Factors of 1067: 1, 11, 97, 1067
  • Factor pairs: 1067 = 1 × 1067 or 11 × 97
  • 1067 has no square factors that allow its square root to be simplified. √1067 ≈ 32.66497

1067 is the hypotenuse of a Pythagorean triple:
715-792-1067 which is 11 times (65-72-97)
We can use the 11 divisibility trick on all the numbers in that triple:
7 – 1 + 5 = 11
7 – 9 + 2 = 0
1 – 0 + 6 – 7 = 0
to see that all of them can indeed be evenly divided by 11.

1067 is palindrome 1F1 in BASE 26 (F is 15 base 10) because 26² + 15(26) + 1 = 1067