Question

What are the smallest and largest four-digit numbers that are:
a) multiples of 9?
b) multiples of 29?
c) multiples of both 9 and 29?

Answer 1

smallest of 9 is 9 it self
snallest of 29 is 29 it self
for both 261

and cant say about largest as no. do not have end

Answer 2

Here is a full solution of the problem:

Let
(a,b,c,29−a−b−c)
(a,b,c,29−a−b−c)
be the decimal representation of such a number, and write ≡≡ for equivalence modulo the prime 2929. We want 999a+99b+9c≡0999a+99b+9c≡0, or
13a+12b+9c≡0 .(1)
(1)13a+12b+9c≡0 .
Put 4b+3c=:r4b+3c=:r. Then (1)(1) implies 13a+3r≡013a+3r≡0, which is satisfied by a=2a=2, r=1r=1, hence enforces
a≡2r≡8b+6c .(2)
(2)a≡2r≡8b+6c .
We now have to determine the solutions of (2)(2) that in addition satisfy the constraints
a∈[1..9],b,c∈[0..9],20≤a+b+c≤27 .(3)
(3)a∈[1..9],b,c∈[0..9],20≤a+b+c≤27 .
This implies b+c≥11b+c≥11, hence b≥2b≥2, so that we can conclude that
70≤8b+6c≤126 .
70≤8b+6c≤126 .
There are only the multiples 3⋅29=873⋅29=87 and 4⋅29=1164⋅29=116 near this range. In view of (2)(2) we therefore have to find the pairs (b,c)(b,c) with
88≤8b+6c≤96,resp.,118≤8b+6c≤124(4)
(4)88≤8b+6c≤96,resp.,118≤8b+6c≤124
that lead to an aa such that the last constraint (3)(3) is satisfied. The first of the ranges (4)(4) contains the pairs
(9,4), (9,3), (8,5), (8,4), (7,6), (6,8), (6,7), (5,9), (5,8) ,
(9,4), (9,3), (8,5), (8,4), (7,6), (6,8), (6,7), (5,9), (5,8) ,
and the second range contains the pairs
(9,8), (8,9) .
(9,8), (8,9) .
For each of these pairs (b,c)(b,c) we now compute aa by means of (2)(2), and retain the triples (a,b,c)(a,b,c) for which the last condition (3)(3) is satisfied. These are the triples
(9,9,4), (7,8,5), (9,6,8), (7,5,9), (4,9,8) .
(9,9,4), (7,8,5), (9,6,8), (7,5,9), (4,9,8) .
From these triples we then obtain the five numbers
9947, 7859, 9686, 7598, 4988
9947, 7859, 9686, 7598, 4988
fulfilling the given conditions.

Hope it helps!!!!!