HOW MANY SETS (A,B,C) OF THREE DIFFERENT,POSITIVE SINGLE DIGIT INTEGERS HAVE THE PROPERTY THAT A SQUARED + B SQUARED + C SQUARED IS A PERFECT SQUARE? please help me with this one a soon as possible its very important and i don't even know where to start!!
well i get that but i just don't c what the answer is becuase it takes forever to do this problem u have to use numbers ubtil you have them all. i am in mathcounts and it is totally competition i cant be up doing a problem like that isnt there a possible way to do it faster and this time maybe state the answer so i can work it backwards ;)
To solve this problem, we need to find all possible sets (A, B, C) of three different positive single-digit integers such that A^2 + B^2 + C^2 is a perfect square.
Here's a step-by-step approach to solve this:
Step 1: List all the single-digit integers.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Step 2: Find all possible combinations of three different integers from the list A.
Step 3: For each combination (A, B, C), calculate A^2 + B^2 + C^2.
Step 4: Check if the result obtained is a perfect square.
Step 5: Finally, count the number of sets (A, B, C) that satisfy the condition.
Let's go through each step in detail:
Step 1: List all the single-digit integers.
A = {1, 2, 3, 4, 5, 6, 7, 8, 9}
Step 2: Find all possible combinations of three different integers from the list A.
This can be done by selecting three integers at a time without repetition. We can use combinatorics to find the combinations.
Step 3: For each combination (A, B, C), calculate A^2 + B^2 + C^2.
Calculate the sum of squares for each combination.
Step 4: Check if the result obtained is a perfect square.
We need to check if the sum of squares is a perfect square. A perfect square is a number that can be obtained by squaring an integer.
Step 5: Count the number of sets (A, B, C) that satisfy the condition.
Keep track of the sets that satisfy the condition, i.e., A^2 + B^2 + C^2 is a perfect square.
By following these steps, you can find the number of sets (A, B, C) that satisfy the given condition.
make 3 columns
A -- B ---C
try
1 1 1---- 1^2 + 1^2 + 1^2 = 3
have a list of perfect squares handy :
1 4 9 16 25 ... 225
is 3 in the list? NO
1 1 2 --- 6 , is 6 in the list ? NO
1 1 3 --- 11 , is 11 in the list ? NO
...
1 1 9 ---83, is 83 in the list ? NO
1 2 1 ---- already done
1 2 2 --- 9, is 9 in the list ? YEAHHHH
1 2 3 ---- 14 NO
1 2 4 --- 21 NO
etc
rather tedious, but do you get the idea?
there are many duplications
e.g. 4 3 2 came up in 2 3 4