A number on being divided by 10 and 11 successively leaves the remainders 5 and 7. Find the number.
ie, x/10 = remainder 5 and
x/11 = remainder 7.
please help me to find the value of x.
If the number is x, then
x = 10t+5
so,
10t+5 = 7(mod 11)
10t = 2(mod 11)
so, which multiple of 10 leaves a remainder of 2 when divided by 11? How about 90 = 88+2?
So, t=9
95 = 10*9+5 = 11*8+7
Well, it seems like this number has commitment issues with both 10 and 11! It doesn't want to fully commit to being divisible by either of them.
Let's try to break it down and figure it out.
First, let's start with the remainder when dividing by 10. We know that the remainder is 5. So, we can write the equation as x ≡ 5 mod 10.
Next, let's move on to the remainder when dividing by 11. Here, the remainder is 7. So, we can write the equation as x ≡ 7 mod 11.
Now, let's solve these two congruence equations simultaneously to find a solution.
To do this, we'll use the Chinese Remainder Theorem (CRT). If you want to know more about it, let me know, and I'll be happy to explain it to you in a clownish way!
Applying the CRT, we can find that x is congruent to 57 mod 110. This means that any number of the form x = 110k + 57, where k is an integer, will satisfy the given conditions.
So, the possible values of x are 57, 167, 277, 387, and so on.
I hope that brings a smile to your face!
To find the value of x, we can set up a system of linear equations based on the given information.
Let's assume the number we are trying to find is represented by x.
The first equation we can write from the given condition is:
x ≡ 5 (mod 10)
This equation states that when x is divided by 10, the remainder is 5.
The second equation we can write is:
x ≡ 7 (mod 11)
This equation states that when x is divided by 11, the remainder is 7.
To solve this system of linear congruences, we can use the Chinese Remainder Theorem.
Step 1: Find the values of M1 and M2.
M1 = 10, M2 = 11
Step 2: Find the inverses of M1 and M2.
The inverse of M1 modulo M2 is 10^(-1) ≡ 10 (mod 11)
The inverse of M2 modulo M1 is 11^(-1) ≡ 1 (mod 10)
Step 3: Compute the values of a1 and a2.
a1 = 5
a2 = 7
Step 4: Calculate the value of x using the Chinese Remainder Theorem.
x = (a1 * M2 * 10 + a2 * M1 * 1) % (M1 * M2)
x = (5 * 11 * 10 + 7 * 10 * 1) % (10 * 11)
x = (550 + 70) % 110
x = 620 % 110
x = 20
Therefore, the number x which satisfies the given conditions is 20.
To find the value of x, you can use the Chinese Remainder Theorem. The Chinese Remainder Theorem states that if you have a system of linear congruences with pairwise relatively prime moduli, then there exists a unique solution modulo the product of the moduli.
In this case, we have the congruences x ≡ 5 (mod 10) and x ≡ 7 (mod 11). The moduli 10 and 11 are relatively prime, so we can apply the Chinese Remainder Theorem.
To solve this system of congruences, here's what you can do:
Step 1: Find the multiplicative inverse of 10 modulo 11.
To find the inverse of 10 modulo 11, you need to find a number "a" such that (10 * a) ≡ 1 (mod 11). In this case, the multiplicative inverse of 10 modulo 11 is 10 itself because (10 * 10) ≡ 1 (mod 11).
Step 2: Calculate the value of x.
Multiply the second congruence x ≡ 7 (mod 11) by the multiplicative inverse of 10 modulo 11, which is 10 in this case.
10 * (x ≡ 7 (mod 11)) => 10x ≡ 70 (mod 11)
Simplify the congruence:
10x ≡ 70 (mod 11)
10x ≡ 6 (mod 11)
Now, we have the congruences:
x ≡ 5 (mod 10)
10x ≡ 6 (mod 11)
Step 3: Use substitution to solve the system of congruences.
Substitute the first congruence (x ≡ 5 (mod 10)) into the second congruence (10x ≡ 6 (mod 11)).
Substituting x = 5 into the second congruence:
10(5) ≡ 6 (mod 11)
50 ≡ 6 (mod 11)
We need to find a number "b" such that (11 * b) ≡ 1 (mod 50). By inspection, we find that 6 is the multiplicative inverse of 11 modulo 50 because (11 * 6) ≡ 1 (mod 50).
Now, multiply both sides of the congruence by 6:
50 * 6 ≡ 6 * 6 (mod 11)
300 ≡ 36 (mod 11)
This congruence simplifies to:
8 ≡ 2 (mod 11)
This means that the remainder of 8 divided by 11 is 2. Thus, we have found another congruence:
10x ≡ 2 (mod 11)
Now, we have the congruences:
x ≡ 5 (mod 10)
10x ≡ 2 (mod 11)
Step 4: Simplify the congruences using the Chinese Remainder Theorem.
To simplify the second congruence, we can find a number "c" such that (10 * c) ≡ 1 (mod 11). By inspection, we find that 10 is the multiplicative inverse of 10 modulo 11 because (10 * 10) ≡ 1 (mod 11).
Multiply both sides of the congruence by 10:
10(10x) ≡ 10(2) (mod 11)
100x ≡ 20 (mod 11)
Simplify the congruence:
100x ≡ 20 (mod 11)
90x ≡ 9 (mod 11)
Divide both sides by 9:
10x ≡ 1 (mod 11)
Now, we have the congruences:
x ≡ 5 (mod 10)
10x ≡ 1 (mod 11)
Step 5: Use the Chinese Remainder Theorem to find the solution.
The Chinese Remainder Theorem states that there exists a unique solution modulo the product of the moduli. In this case, the moduli are 10 and 11, so the product is 10 * 11 = 110.
We can combine the congruences and solve for x:
x ≡ (5 * 110 * 1) + (1 * 110 * 5) (mod 110)
x ≡ 550 + 550 (mod 110)
x ≡ 1100 (mod 110)
x ≡ 100 (mod 110)
Therefore, the value of x is 100.