000RM.dvi

410 Digit problems

Exercise

1.Lucky Larry, a mathematics student whose plausible mistakes in

computation always results in correct answers, once wrote an an-

swer in the form

abca=abca

whereabcarepresents a four-digit integer whose digitsa,b,care

all different. What specific number did Lucky Larry write?

2.Find all natural numbers whose square (in base 10) is represented

by odd digits only.

3.LetN be the sum of the digits of a natural numberA, letB =

A+N, letA′be the sum of the digits of the numberB, and let

C =B+A′. FindAif the digits ofCare those ofAin reverse

order.

Solution. IfAhaskdigits, thenCcannot exceedAby more than

18 k. On the other hand, ifCis the reverse ofA, thenCexceedsA

by at least 9 · 10

k− 2

(^2) iskis even, and by at least 99 · 10
k− 3
(^2) ifkis
odd. This quickly impliesk≤ 2. From this, we findA=12or 69.
4.Find the three 3-digit numbers each of which is equal to the product
of the sum of its digits by the sum of the squares of its digits.
5.Find all 4-digit numbersabcdsuch that^3

√

abcd=a+b+c+d.

6.Use each digit 1, 2, 3, 4, 5, 6, 7, 8, 9 exactly once to form prime

numbers whose sum is smallest possible.

What if we also include the digit 0?

7.There are exactly four 3-digit numbers each equal to the sum of the

cubes of its own digits. Three of them are 153, 371, and 407. What

is the remaining one?

8.Find digitsm,a,b,c,d,e,fsuch that

abcdef

fedcba

=

9 m

9 m+1

.

9.Find a number of the formaaabbbccc, which when increased by 1,

gives a square.