Example 1. The numbers 46 and 96 are rather peculiar; their product does not change if the digits are interchanged. Have a look: 46 × 96 = 4416 = 64 × 69. Find all two-digit numbers having the same property.
Example 2. How many times the digits of a computer keyboard are required to press in typing 1st 100 natural numbers?
Of course, you can find the solution just below, but it is highly recommended that you first try to solve it on your own.
Just remember the words of Paul Halmos, who says “the only way to learn mathematics is to do mathematics”.
Solution 1. Denoting the digits of the desired numbers by x and y, z and t, where x, y, z, t are integers less than 10, we set up the equation
(10x + y) (l0z + t) = (10y + x)(10t + z)
Or, 100xz + 10xt + 10yz + yt = 100yt + 10yz + 10xt + xz
Or, 99xz = 99yt
Or, xz = yt.
To find the solutions we set up pairs of equal products made up of 9 digits:
- 1 × 4 = 2 × 2 2 × 8 = 4 × 4 1 × 6 = 2 × 3
- 2 × 9 = 3 × 6 1 × 8 = 2 × 4 3 × 8 = 4 × 6
- 1 × 9 = 3 × 3 4 × 9 = 6 × 6 2 × 6 = 3 × 4
There are nine equalities. From each one, it is possible to set up one or two desired groups of numbers. For example,
using the equality 1 × 4 = 2 × 2, we find one solution: 12 × 42 = 21 × 24;
using 1 × 6 = 2 × 3 we get two solutions: 12 × 63 = 21 × 36, 13 × 62 = 31 × 26.
In this manner we obtain the following 14 solutions:
- 12 × 42 = 21 × 24 23 × 96 = 32 × 69
- 12 × 63 = 21 × 36 24 × 63 = 42 × 36
- 12 × 84 = 21 × 48 24 × 84 = 42 × 48
- 13 × 62 = 31 × 26 26 × 93 = 62 × 39
- 13 × 93 = 31 × 39 34 × 86 = 43 × 68
- 14 × 82 = 41 × 28 36 × 84 = 63 × 48
- 23 × 64 = 32 × 46 46 × 96 = 64 × 69
Solution 2. Following observations are important:
- 1. Numbers can be of single digit, two digits or three digits.
- 2. Number of different digit numbers (1, 2 or 3) are different.
Number of single-digit natural numbers is 9 (namely, 1–9).
So, the number of times the digits of a computer keyboard are required to press = 9 × 1 = 9.
Number of two-digit natural numbers is 90 (namely, 10–99).
So, the number of times the digits of a computer keyboard are required to press = 90 × 2 = 180.
Number of three-digit natural numbers is 1 (namely, 100).
So, the number of times the digits of a computer keyboard are required to press = 1 × 3 = 3.
Hence, the total number of times the digits of a computer keyboard are required to press = 9 + 180 + 3 = 192.
Note. It can be generalized now that number of four-digit natural numbers = 9000 and so on.