1st Irish 1988 Problem 9 The year 1978 had the property that 19 + 78 = 97. In other words the sum of the number formed by the first two digits and the number formed by the last two digits equals the number formed by the middle two digits. Find the closest years either side of 1978 with the same property.
1868, 2307. Solution Suppose the number is abcd. So we have 10a + b + 10c + d = 10b + c, or 9(b - c) = 10a + d. So 10a + d is a multiple of 9. So it must be 18, 27, 36, 45, 54, 63, 72, 81, 90 or 99. Thus if a = 1, then d must be 8 and b = c + 2. Thus the closest such number below 1978 is 1868. The closest number above must have a = 2, d = 7, and b = c + 3. We want b as small as possible, so we take b = 3, c = 0, giving 2307. 1st Irish 1988 © John Scholes jscholes@kalva.demon.co.uk 11 June 2003 Last updated/corrected 11 Jun 03