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Pythagoras, when asked what a friend was, replied: "One who is the other I, such are 220 and 284."
Expressed in modern terminology this meant:
The divisors of 284 are 1, 2, 4, 71 and 142 and these add up to 220.
The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110 and these in turn add up to 284.
Such numbers the Pythagoreans called amicable numbers.
(W.H. Auden : A Certain World)
Then there were the perfect numbers. Consider first a number such as 14. Add up its divisors (1, 2 and 7). We get 10. The number 14 is greater than the sum of its divisors and is for this reason called excessive.
On the other hand, the sum of the divisors of 12 (1, 2, 3, 4, 6) is 16 which is greater than 12 and for this reason 12 is said to be defective.
But in a perfect number there is neither excess nor deficiency. The number equals the sum of its own divisors. The smallest perfect numbers are 6 and 28.
Name an 18-digit number in which no two adjacent digits are alike and is four times its reversal.
(A) 871208712871 208712
(B) 871 287912879128712
Prof. G.H. Hardy visited a nursing home in Cambridge where the mathematical genius Ramanujan was ailing. He mentioned that he arrived by a taxi bearing the number 1729. Ramanujan immediately observed - it is the smallest possible integer [the least of the numbers : a whole number] that can be expressed as a sum of two cubes in two different ways. This number (1729) came to be ever associated with Ramanujan.
(Word of The Mouth)
1729 = 103 + 93
1729 = 123 + 13