Prize Puzzle Competition 22

Closing date is 30th April 2000

64. A 'Harshad number' has been defined as a number which can be divided exactly by the sum of its digits.
For example: 1729 is a Harshad number since 1 + 7 + 2 + 9 = 19 and 1729 can be divided exactly by 19.

Find the smallest Harshad number which is divisible by 13

(Harshad numbers were defined by D R Kaprekar (a mathematician from India) who also gave us
'Kaprekar's Constant'. The word 'Harshad' comes from the Sanskrit language and means 'great joy'.)

65. Each of the letters A, B, C, D, E is used to represent a single
digit in these two statements.(Same letter = same digit.)

A + B = C
C + D = EA
(Note EA is a 2-digit number)

What is the value of B + D ?

66. Two planes fly once around the world together.
They both fly at a constant distance above the
surface of the Earth, one above the other.
On completion, one of the planes has travelled
two kilometres further than the other.
(Assume the Earth is a perfect sphere.)

How far apart (in metres) were the two planes?

The winner of Competition #21 was Caroline Lines of Cardiff
The answers were:

62.22 + 2 or 3^3 - 3 or 9 x sqrt(9) - sqrt(9) or 4 + 4! - 4
63. 1683 different ways.