Today’s Puzzle:

Ten years ago today, I published my first blog post. I decided to commemorate the day by making a Factor Fits puzzle with not one, but two, tens in it.

Factors of 1767:

This is my 1767th post. What are the factors of 1767?

• 1767 is a composite number.
• Prime factorization: 1767 = 3 × 19 × 31.
• 1767 has no exponents greater than 1 in its prime factorization, so √1767 cannot be simplified.
• The exponents in the prime factorization are 1, 1, and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 = 8. Therefore 1767 has exactly 8 factors.
• The factors of 1767 are outlined with their factor pair partners in the graphic below.

More About the Number 1767:

1767 is the difference of two squares in four different ways:
884² – 883² = 1767,
296² – 293² = 1767,
56² – 37² = 1767, and
44² – 13² = 1767.

1767 is the sum of consecutive numbers in several different ways. In each case, I’ve highlighted the numbers in the middle of the sum. What do you notice about those numbers?

1767 is the sum of two consecutive numbers:
883 + 884 = 1767.

It is the sum of three consecutive numbers:
588 + 589 +590 = 1767.

1767 is the sum of six consecutive numbers:
292 + 293 + 294 + 295 + 296 + 297 = 1767

It is the sum of 19 consecutive numbers:
84 + 85 + 86 + 87 + 88 + 89 + 90 + 91 + 92 + 93 + 94 + 95 + 96 + 97 + 98 + 99+ 100 + 101 + 102 = 1767.

1767 is the sum of 31 consecutive numbers:
42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 + 72 = 1767.

It is the sum of 38 consecutive numbers:
28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 = 1767

AND 1767 is the sum of 57 consecutive numbers!
3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 1767.

1767 is repdigit, VV, in base 56 because
V(56) + V(1) = 1767. What does V equal? You can figure it out!

V(56) + V(1) = 1767.
V(57) = 1767.
V = 1767/57.