Ernst Kummer's 7*9 anecdote

Prussian born Ernst Eduard Kummer (January 29, 1810 – May 14, 1893) was one of the greatest mathematicians ever lived.

Despite being called the father of modern arithmetic (that is, number theory), Kummer was rather poor at simple arithmetic. Once, in a class, he needed to find the product of seven and nine. “Seven times nine,” he began, “Seven times nine is er – ah --- ah --- seven times nine is ….” “Sixty-one,” a student suggested. Kummer wrote 61 on the board. “Sir,” said another student, “it should be sixty-nine.” “Come, come, gentlemen, it can’t be both,” Kummer exclaimed. “It must be one or the other.” Kummer then calculated 7 x 9 as follows: “Hmmm the product cannot be 61, because 61 is prime, it cannot be 65, because 65 is a multiple of 5, 67 is a prime, 69 is too big. Only 63 is left.”

Magic Number 1001

Select any three digit number (say 176). Now make a six digit number by repeating the selected number (i.e. in our case 176176).
Now divide the number with 13 then with 11 and then with 7. Answer will be the original three digit number. (i.e. 176)

The math behind the fact:
By repeating the number besides the original, we are actually doing a 1001 multiplication.
abc(1001) = abc (100 + 1) = abc000 + abc = abcabc.
And by doing a division of 13, followed by 11, and 7, we are trying to do a division of 1001.Therefore abc (1001) / 1001 gives us back the original abc.

Magic Number 1089

Write down a three-digit number whose digits are decreasing. Then reverse the digits to create a new number, and subtract this number from the original number. With the resulting number, add it to the reverse of itself. The number you will get is 1089!

For example, if you start with 532 (three digits, decreasing order), then the reverse is 235. Subtract 532-235 to get 297. Now add 297 and its reverse 792, and you will get 1089!

The math behind the fact:

If we let a, b, c denote the three digits of the original number, then the three-digit number is 100a+10b+c. The reverse is 100c+10b+a. Subtract: (100a+10b+c)-(100c+10b+a) to get 99(a-c). Since the digits were decreasing, (a-c) is at least 2 and no greater than 9, so the result must be one of 198, 297, 396, 495, 594, 693, 792, or 891. When you add any one of those numbers to the reverse of itself, you get 1089!