Chapter 19

More digital trivia

1.Larry, Curly, and Moe had an unusual combination of ages. The

sum of any two of the three ages was the reverse of the third age

(e.g.,16 + 52 = 68, the reverse of 86). All were under 100 years

old.

(a) What was the sum of the ages?

(b) If Larry was older than either of the others, what was the youngest

he could be?

2.Letnbe a nonnegative integer. The number formed by placing 2 n

and 2 n+1side by side in any order is divisible by 3.^1

3.Find positive integersx, y, z(less than 100), such thatx^2 +y^2 =

z^2 andX^2 +Y^2 =Z^2 whereX, Y, Zare derived fromx, y, zby

inserting an extra digit (the same for all) on the left.

4.(a) Find the smallest positive integerNhaving the property that the

sum of its digits does not divide the sum of the cubes of its digits.

(b) Find the two consecutive positive integers each of which equals

the sum of the cubes of its digits.^2

5.Find two perfect cubes which, considered jointly, contain the digits

0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 each once. Is the solution unique?^3

6.Show that there is but one five-digit integer whose last three digits

are alike and whose square contains no duplicate digits.^4

(^1) Suppose 2 nhaskdigits. Putting 2 n+1on the left hand side of 2 ngives the number 2 n+1· 10 k+2n.
Modulo 3, this is(−1)n+1+(−1)n≡ 0.
2 ; (b) 370 and 371.=112N(a)
(^3) Unique solution:. = 804357^393 and = 9261^321
4. = 6597013284^281222