the remainder is 9. Take 63, reverse the digits, and subtract 36 from 63; you have 27, a multiple of 9, and 2 + 7 = 9. Once more, the number 13 is the reverse of 31; the difference between these numbers is 18, or twice 9.
Again, the same property found in two numbers thus changed, is discovered in the same numbers raised to any power.
Take 21 and 12 again. The square of 21 is 441, and the square of 12 is 144; subtract 144 from 441, and the remainder is 297, a multiple of 9; besides, the digits expressing these powers added together give 9. The cube of 21 is 9261, and that of 12 is 1728; their difference is 7533, also a multiple of 9.
The number 37 has also somewhat remarkable properties; when multiplied by 3 or a multiple of 3 up to 27, it gives in the product three digits exactly similar. From the knowledge of this the multiplication of 37 is greatly facilitated, the method to be adopted being to multiply merely the first cipher of the multiplicand, by the first of the multiplier; it is then unnecessary to proceed with the multiplication, it being sufficient to write twice to the right hand the cipher obtained, so that the same digit will stand in the unit, tens, and hundreds places.
