A question about the natural number N:?

First, where you said 0 ≤ a,b ≤ *b* I assume you meant 0 ≤ a,b ≤ 9.

There are several ways to approach this. Here's just one method.

We can think of N as:

162080 + 100a + b

When we divide 162080 by 7, the remainder is 2. So the following expression will give the same remainder when divided by 7.

100a + b + 2

Also, since 98 is a multiple of 7, we can drop 98a. This will also give the same remainder when divided by 7.

2a + b + 2

Now we can easily plug in a = 0 to 9 and calculate the possible values of b where the remainder is 0.

a = 0 --> b = 5

a = 1 --> b = 3

a = 2 --> b = 1 or 8

a = 3 --> b = 6

a = 4 --> b = 4

a = 5 --> b = 2 or 9

a = 6 --> b = 0 or 7

a = 7 --> b = 5

a = 8 --> b = 3

a = 9 --> b = 1 or 8

Answers:

162085

162183

162281

162288

162386

162484

162582

162589

162680

162687

162785

162883

162981

162988

P.S. Another way is brute force all 100 choices of a (0-9) and b(0-9) and test them for divisibility by 7. I did this also and it concurs with my answers above.

n=162a8b=162080+100a+b, 0=<a,b<=9

If b is removed, then n->n'=162a8. Let

16208+10a-2b=0(mod 7), then

10a=(2b-16208)(mod 7)-----(1)

Now try b=0,1,2,...,9 in turn in (1) & find "a" accordingly.

The results are as listed:

| b |-----0-----|-----1-----|-----1-----|-----2-----|-----3-----|-----3-----|

| a |-----6-----|-----2-----|-----9-----|-----5-----|-----1-----|-----8-----|

| n | 162680 | 162281 | 162981 | 162582 | 162183 | 165883 |

| b |-----4-----|-----5-----|-----5-----|-----6-----|-----7-----|-----8-----|

| a |-----4-----|-----0-----|-----7-----|-----3-----|-----6-----|-----2-----|

| n | 162484 | 162085 | 162785 | 162386 | 162687 | 162288 |

| b |-----8-----|-----9-----|

| a |-----9-----|-----5-----|

| n | 162988 | 162589 |

For examples,

(i) From (1), input b=3, then

10a=-16202(mod 7)=>

10a=3(mod 7)=>

a=1(mod 7)=>

a=1+7k=>

a=1, k=0

a=8, k=1

=>

n=162183 or n=165883

(ii) input b=5, then

10a=-16198(mod 7)=>

10a=0(mod 7)=>

a=0+7k=>

a=0, k=0

a=7, k=1

=>

n=162085 or n=162785

etc.

If using the method of "trial & error", we

need to try at least 10*10=100 times.

You mistyped this, where you wrote b<=b.  Maybe you meant b<=9 ?

This is a pretty simple problem, you can start at 162085 (which is a multiple of 7) and just keep adding either 7 or 98, depending on whether adding the 7 makes the "8" turn into a "9".

For instance 162085 + 7 is too much, so you go next to

162085 + 98 = 162183.

Then 162183 + 7 is too much, so you next go to

162183 + 98 = 162281.

Then 162281 + 7 is okay, your next success is

162288.

Then 162288 + 98 = 162386.

et cetera!

Pretty soon you get above 163000 so you have to stop.

Every seven numbers, one of them is divisible by 7. So consider

162080.

Which is the first number divisible by 7? [ you only have the seven first to try].

Answer that, then we will take it from there...