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.