Chaining up two natural numbers (concatenation)

Introduction: A mathematical transformation be defined by the fact that two natural numbers are "concatenated" to a third natural number:
n = 10m * a + b, where m = [lg b] + 1 . ([x] = largest integer less than or equal to x)
Two examples of a concatenation:
45, 386 ⇒ 45386
7, 78230 ⇒ 778230
Note: The number of digits of a natural number u be understood as writing u without leading zeros.

The now following problem is an arithmetical problem for the defined facts:

Given: A three-digit number b is the twenty-nine times of a two-digit number a.
a and b are to be concatenated to "their" five-digit number n (the two-digit number forms the left part of n).

Wanted: A proof for the following facts:

(1) There is more than one solution.
(2) All solutions are divisible by 7³ without remainder. 

Another problem on the topic of "concatenation"

A concatenation of two natural numbers a and b leads to n times the value of a and m times the value of b.

p = [lg (b)] + 1

a * 10[lg (b)]+1 + b = m * b
a * 10[lg (b)]+1 + b = n * a

a * 10p + b = m * b
a * 10p + b = n * a

⇒ m * b = n * a

Example: The numbers 24 and 600 are chained. The result is the number 24600, and the value of the result of this transformation can be expressed as 1025 times 24 on the one hand and 41 times 600 on the other:

1025 * 24 = 41 * 600 = 24600 = v (24, 600) =
= 24 * 10[lg (600)]+1 + 600

Practice examples:

(1) The result of chaining two numbers is 1008 times the first component a and 126 times the second component b. At least one solution pair should be found.
(2) The result of a concatenation of two numbers is 1020 times the first component a and 51 times the second component b. At least one solution pair shall be found.

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