This is an elementary number theory problem for college mathematics majors. In ancient China, it was called Sun Tzu's algorithm, and it was also called Han Xin's point soldier problem.

Your algorithm for this problem is:

Step 1: 3× 5×11=165.

Step 2: 3× 5 =15×11= 55×1= 33.

Step 3:15× 3 = 4555×1= 5533× 2 = 66.

Because 45 is divided by 1 1, 55 is divided by 3 = 1 and 66 is divided by 5 = 1.

Step 4: 45×3+55× 1+66×2=322.

So 322 meets the problem.

Step 5: 322-165 = 157157 is less than165, then157 is the smallest positive integer that satisfies the meaning of the question.

Example 1: A number divided by 3 is 1, divided by 4 is 2, and divided by 5 is 4. What's the smallest number?

The numbers 3, 4 and 5 in the question are pairwise coprime.

Then [4,5] = 20; 〔3,5〕= 15; 〔3,4〕= 12; 〔3,4,5〕=60。

To divide 20 by 3 to get 1, use 20× 2 = 40;

Divide 15 by 4 to get 1, and use15× 3 = 45;

Divide 12 by 5 to get 1, and use 12×3=36.

Then, 40× 1+45× 2+36× 4 = 274,

Because, 274 >; 60, so, 274-60× 4 = 34, which is the number to be found.

Example 2: What is the smallest number when a number is divided by 3, 4 by 7 and 5 by 8?

3, 7 and 8 in the problem are pairwise coprime.

Then [7,8] = 56; 〔3,8〕=24; 〔3,7〕=2 1; 〔3,7,8〕= 168。

In order to divide 56 by 3 to get 1, use 56× 2 =112;

Divide 24 by 7 to get 1, and use 24×5= 120.

Divide 2 1 by 8 to get 1, and use 21× 5 =105;

Then112× 2+120× 4+105× 5 =1229,

Because,1229 >; 168 Therefore, 1229- 168× 7 = 53, which is the number to be found.

Example 3: Divide a number by 5+4, 8+3, 1 1+2 to find the minimum natural number that meets the conditions.

The numbers 5,8, 1 1 in the problem are pairwise coprime.

Then [8,11] = 88; 〔5, 1 1〕=55; 〔5,8〕=40; 〔5,8, 1 1〕=440。

In order to divide 88 by 5 to get 1, use 88× 2 =176;

Divide 55 by 8 to get 1, 55× 7 = 385;

Divide 40 by 1 1 and use 40×8=320.

Then, 176× 4+385× 3+320× 2 = 2499,

Because, 2499 >; 440, so, 2499-440× 5 = 299, which is the number of seeking.

Example 4: There is a classmate in a certain grade, with five students for every nine people, one for every seven people 1 student, and two for every five people. How many students are there at least in this grade?

The numbers 9, 7 and 5 in the question are pairwise coprime.

Then [7,5] = 35; 〔9,5〕=45; 〔9,7〕=63; 〔9,7,5〕=3 15。

In order to divide 35 by 9 to get 1, use 35× 8 = 280;

Divide 45 by 7 to get 1, 45× 5 = 225;

Divide 63 by 5 to get 1, and use 63×2= 126.

Then, 280× 5+225×1+126× 2 =1877,

Because,1877 >; 3 15 Therefore, 1877-3 15× 5 = 302, which is the number to be found.

Example 5: There is a classmate in a certain grade, with 6 people in a row for every 9 people, 2 people in a row for every 7 people and 3 people in a row for every 5 people. How many people are there at least in this grade?

The numbers 9, 7 and 5 in the question are pairwise coprime.

Then [7,5] = 35; 〔9,5〕=45; 〔9,7〕=63; 〔9,7,5〕=3 15。

In order to divide 35 by 9 to get 1, use 35× 8 = 280;

Divide 45 by 7 to get 1, 45× 5 = 225;

Divide 63 by 5 to get 1, and use 63×2= 126.

Then, 280× 6+225× 2+ 126× 3 = 2508,

Because, 2508 >; 3 15, so 2508-3 15× 7 = 303, is the number to be found.

T0928 and king__dom both use the knowledge of congruence in number theory, which is relatively easy.