Repunits are whole numbers that consist of only ones.
Repunits have a few patterns when they are multiplied (up to 8 digits).
Pattern 1
The product always ends in 1.
Example:
1x1=1
11x11=121
111x111=12,321
1,111x1,111=1,234,321
11,111x11,111=123,454,321
111,111x111,111=12,345,654,321
1,111,111x1,111,111=1,234,567,654,321
11,111,111x11,111,111=123,456,787,654,321
Pattern 2
The product always has an odd number of digits. Each product increases by two.
Example:
1x1 product is one digit, 11x11 product is three digits, and 111x111 product is five digits, 1,111x1,111 product is seven digits, and 11,111x11,111 product is nine digits, and so on.
1x1=1----> 1 digits
11x11=121----> 3 digits
111x111=12,321----> 5 digits
1,111x1,111=1,234,321----> 7 digits
11,111x11,111=123,454,321----> 9 digits
111,111x111,111=12,345,654,321----> 11 digits
1,111,111x1,111,111=1,234,567,654,321----> 13 digits
11,111,111x11,111,111=123,456,787,654,321----> 15 digits
Pattern 3
The numbers in the product go in ascending order(going up) up to the number of digits in the factors, then go in descending order(going down) minus one less digit than the factors. Also, the first number in descending order is one less than the last number in ascending order.
Example:
11x11
Factor 11 has two digits. Product 121 has three digits. The product has two numbers going in ascending order, then one number going in descending order. Number 1 is the first number going in descending order. Number 1 is one less than number 2 which the last number in ascending order.
11x11=121----> Goes up 1,2 then goes down 1.
111x111
Factor 111 has three digits. Product 12,321 has five digits. The product has three numbers going in ascending order, then two numbers going in descending order. Number 2 is the first number going in descending order. Number 2 is one less than number 3 which the last number in ascending order.
111x111=12,321---->Goes up 1,2,3, then goes down 2,1.
1,111x1,111
Factor 1,111 has four digits. Product 1,234,321 has seven digits. The product has four numbers going in ascending order, then three numbers going in descending order. Number 3 is the first number going in descending order. Number 3 one less than number 4 which the last number in ascending order.
1,111x1,111=1,234,321----> Goes up 1,2,3,4 then goes down to 3,2,1.
11,111x11,111
Factor 11,111 is five digits. Product 123,454,321 is nine digits. The product has five numbers going in ascending order, then four numbers going in descending order. Number 4 is the first number going in descending order. Number 4 is one less than number 5 which the last number in ascending order.
11,111x11,111=123,454,321----> Goes up 1,2,3,4,5 then goes down 4, 3,2,1.
111,111x111,111
Factor 111,111 has six digits. Product 12,345,654,321 has eleven digits. The product has six numbers going in ascending order, then five numbers going in descending order. Number 5 is the first number going in descending order. Number 5 is one less than number 6 which the last number in ascending order.
111,111x111,111=12,345,654,321----> Goes up 1,2,3,4,5,6 then goes down 5,4,3,2,1.
1,111,111x1,111,111
Factor 1,111,111 has seven digits. Product 1,234,567,654,321 has thirteen digits. The product has seven numbers going in ascending order, then six numbers going in descending order. Number 6 is the first number going in descending order. Number 6 is one less than number 7 which the last number in ascending order.
1,111,111x1,111,111=1,234,567,654,321---->Goes up 1,2,3,4,5,6,7 then goes down 6,5,4,3,2,1.
11,111,111x11,111,111
Factor 11,111,111 has eight digits. Product 123,456,787,654,321 has fifteen digits. The product has eight numbers going in ascending order then seven numbers going in descending order.
11,111,111x11,111,111=123,456,787,654,321----> Goes up 1,2,3,4,5,6,7,8 then goes down 7,6,5,4,3,2,1. Number 7 is the first number going in descending order. Number 7 is one less than number 8 which the last number in ascending order.
Today's fun math fact via Su, Francis E., et al. “Repunit Fun.” Math Fun Facts. <https://www.math.hmc.edu/funfacts>.