Find the largest positive integer that leaves the same remainder when divided into each of 889, 961, 1009, and 1189
889/12 = 74 with remainder 1
961/12 = 80 with remainder 1
1009/12 = 84 with remainder 1
1189/12 = 99 with remainder of 1
So 12 is it!
I am not sure if you know anything about modular arithmetic, so I made up a silly little old-fashioned computer program.
Here is a listing of it, with the output at the end.
The first column of the output is the remainder, the second number is the largest positive integer.
It found none larger than 12
10 for n = 2 to 889
20 r1 = 889-int(889/n)*n
30 r2 = 961-int(961/n)*n
40 if r1 = r2 then 50 else goto 100
50 r3 = 1009-int(1009/n)*n
60 if r3 = r2 then 70 else goto 100
70 r4 = 1189-int(1189/n)*n
80 if r4 = r3 then 90 else goto 100
90 print r4,n
100 next n
>run
1 2
1 3
1 4
1 6
1 12
I did this with something called "Chipmunk Basic"
To find the largest positive integer that leaves the same remainder when divided into each of 889, 961, 1009, and 1189, we can make use of the concept of remainders and common factors.
Step 1: Find the differences between the given numbers
Take the differences between each pair of given numbers:
961 - 889 = 72
1009 - 961 = 48
1189 - 1009 = 180
Step 2: Find the greatest common divisor (GCD) of these differences
The GCD is the largest positive integer that divides each of the given differences. In this case, we need to find the GCD of 72, 48, and 180.
Using the Euclidean algorithm, we can find the GCD of two numbers at a time:
GCD(72, 48) = 24
GCD(24, 180) = 12
So, the GCD of 72, 48, and 180 is 12.
Step 3: Subtract the GCD from any of the given numbers
Subtract the GCD (12) from any of the given numbers (889 in this case):
889 - 12 = 877
Step 4: Verify if the result leaves the same remainder when divided into the other numbers
Check if the result (877) leaves the same remainder when divided into the other numbers (961, 1009, and 1189):
961 % 877 = 84
1009 % 877 = 132
1189 % 877 = 312
Since the remainder is the same for all three divisions, we have found our answer.
Therefore, the largest positive integer that leaves the same remainder when divided into each of 889, 961, 1009, and 1189 is 877.