what is the least common positive integer that meets the following conditions:
divided by 7 with remainder 4
divided by 8 with remainder 5
divided by 9 with remainder 6
i thought you could add 7 and 4 to get 13, then divide 13 and 7 with r=4, but it has to be the same number for all of them. and it's not any numbers between 0 and 215
You can use the Chinese Remainder Theorem. Notation: If a number X divided by N has a remainder of r, we write
X Mod N = r
("Mod" is shorthand for "Modulo")
So, if we call the solution X, we have:
X Mod 7 = 4
X Mod 8 = 5
X Mod 9 = 6
If you are given some number X and you want to compute X mod 7, then you can subtract from X any multiple of 7 without affecting the result. If there exists a number Y such that
X* Y Mod 7 = 1,
then we call Y the inverse of X and vice versa. E.g. 3*5 Mod 7 = 1, so, Mod 7, the inverse of 3 is 5 and the inverse of 5 is 3. Let's use this notation for the inverse:
[3]_7 = inverse of 3 mod 7 = 5
You can now directly write down the solution of your problem as:
X = 4*8*9[8*9]_7 + 5*7*9[7*9]_8 +
6*7*8[7*8]_9
Let's check that this is the solution. If you compute X Mod 7, the first term will yield 4 because 4 is multiplied by 8*9 and then we multiply by the inverse of 8*9 mod 7, so the 8*9 multiplied by the inverse of 8*9 yields 1 Mod 7 and we are left with 4.
What about the other two terms? Well, they are all proportional to 7, so they yield zero Mod 7.
So, we see that X Mod 7 = 4
Similarly, you can see that x Mod 8 = 5. The first and last terms are proportional to 8 and are thus zero Mod 8. The second term is, Mod 8, equal to 5, because it is 5 times 7*9 times the inverse of 7 * 9.
And similarly, it is easy to see that X Mod 9 = 6.
So, to find X you have to work out the inverses. We have:
8*9 Mod 7 = 2
Note that to compute this you can reduce 8 Mod 7 first which is 1 and 9 Mod 7 = 2, so 8*9 Mod 7 =
1*2 Mod 7 = 2.
The inverse of 8*9 Mod 7 is thus the same as the inverse of 2 Mod 7. Now, we have:
2*3 Mod 7 = 6 Mod 7 = -1 Mod 7
So, 2*(-3) Mod 7 = 1 Mod 7
So, the inverse is -3 Mod 7 = 4
Indeed 2*4 = 8 and 8 Mod 7 = 1
The next term we need to calculate is:
[7*9]_8
Now Mod 8 we have that:
7 Mod 8 = -1 Mod 8
9 Mod 8 = 1
So, we need the inverse of -1, which we can take to be -1 or -1 + 8 = 7.
Finally we have to compute:
[7*8]_9
7 Mod 9 = -2 Mod 9
8 Mod 9 = -1 Mod 9
So 7*8 Mod 9 = 2
The inverse of 2 is, Mod 9, equal to 5 because 2*5 = 10 and Mod 9 we have 10 Mod 9 = 1.
The solution is thus:
X = 4*8*9[8*9]_7 + 5*7*9[7*9]_8 +
6*7*8[7*8]_9 =
X = 4*8*9*4 - 5*7*9 +
6*7*8*5 = 2517
By adding a multiple of 7*8*9 to a solution, you get another solution because 7*8*9 is zero Mod 7, Mod 8 and Mod 9. All possible solutions can be obtained this way. The smalles solution is thus:
2517 Mod (7*8*9) = 501
To find the least common positive integer that meets the given conditions, you can approach it using the Chinese Remainder Theorem. The Chinese Remainder Theorem states that if we have a system of congruences as follows:
x ≡ a (mod m)
x ≡ b (mod n)
x ≡ c (mod p)
where a, b, and c are the remainders, and m, n, and p are the divisors, then we can find a unique solution for x.
In this case, we have the following congruences:
x ≡ 4 (mod 7)
x ≡ 5 (mod 8)
x ≡ 6 (mod 9)
To solve this system of congruences, we can use a method called the Chinese Remainder Theorem. Here are the steps:
Step 1: Find the product of the divisors. In this case, m⋅n⋅p = 7⋅8⋅9 = 504.
Step 2: For each congruence, divide the product of the divisors by the respective divisor to get the value of N. In this case, we have N1 = 504/7 = 72, N2 = 504/8 = 63, and N3 = 504/9 = 56.
Step 3: Find the modular multiplicative inverses of N1, N2, and N3 with respect to the respective divisors. Let's denote them as a1, a2, and a3.
For N1: 72 * a1 ≡ 1 (mod 7)
For N2: 63 * a2 ≡ 1 (mod 8)
For N3: 56 * a3 ≡ 1 (mod 9)
Step 4: Calculate the sum of the products of the remainders and the modular multiplicative inverses. Denote this as x.
x = a1 * N1 * 4 + a2 * N2 * 5 + a3 * N3 * 6
Step 5: Reduce x modulo the product of the divisors to get the least positive integer that meets the conditions.
x % 504 = The least common positive integer.
By following these steps, you can calculate the least common positive integer that satisfies the given conditions.