Find the greatest number which when divided by 179 and 234 leaves a remainder of 3 in each case
To find the greatest number that satisfies the given conditions, we can utilize the Chinese Remainder Theorem (CRT).
Since the remainder must be 3 when divided by both 179 and 234, we can write two congruences:
x ≡ 3 (mod 179)
x ≡ 3 (mod 234)
First, let's find the modular inverse of 234 modulo 179. By using the Extended Euclidean Algorithm, we can find that the inverse of 234 modulo 179 is -73.
Next, we can use this inverse to solve the system of congruences. By multiplying the second congruence by -73, we obtain:
-73x ≡ -219 (mod 179)
We can simplify this congruence by adding 179 to both sides:
-73x + 179y ≡ -40 (mod 179)
106x + 179y ≡ -40 (mod 179)
Now we have two congruences:
x ≡ 3 (mod 179)
106x + 179y ≡ -40 (mod 179)
To simplify the second congruence, we can divide both sides by 106:
x ≡ 3 (mod 179)
x + (179/106)y ≡ -40/106 (mod 179)
x + (179/106)y ≡ -20/53 (mod 179)
Since 179 is a prime number, (179/106) is congruent to the modular inverse of 106 modulo 179. We can use the Extended Euclidean Algorithm to find that the inverse of 106 modulo 179 is 115.
Therefore, we have:
x ≡ 3 (mod 179)
x + 115y ≡ -20/53 (mod 179)
Now we can use CRT to solve these congruences:
x ≡ 3 (mod 179)
x ≡ -20/53 - 115y (mod 179)
Since x is the greatest number that satisfies the conditions, it can be written in the form x = 179k + 3, where k is an integer.
Substituting this into the second congruence, we have:
179k + 3 ≡ -20/53 - 115y (mod 179)
To eliminate the fractional term, we can multiply both sides by 53:
179k + 3 ≡ -20 - 6115y (mod 9537)
Simplifying further, we have:
179k ≡ -23 - 6115y (mod 9537)
To solve for the greatest value of k, we need to find the largest negative value for the right-hand side of the congruence. Since the remainder must be positive, we can let y = 0 and find the corresponding value of k:
179k ≡ -23 (mod 9537)
Using the Extended Euclidean Algorithm, we find that the inverse of 179 modulo 9537 is 83. Multiplying both sides of the congruence by 83, we get:
k ≡ -23 * 83 ≡ -1909 (mod 9537)
Therefore, the largest negative value for k is -1909 + 9537 = 7628.
Substituting this value of k back into the expression for x, we have:
x = 179(7628) + 3 = 1368453
Therefore, the greatest number that satisfies the given conditions is 1368453.
To find the greatest number that satisfies both conditions, we need to find the least common multiple (LCM) of the two divisors, 179 and 234, and subtract 3 from it.
Step 1: Find the LCM of 179 and 234.
We can use the prime factorization method to find the LCM.
Prime factorization of 179:
179 = 1 × 179 (179 is a prime number)
Prime factorization of 234:
234 = 2 × 3 × 3 × 13
Now we write down the factors of both numbers, multiplied by each prime factor the greatest number of times it appears:
179 = 1 × 179
234 = 2 × 3 × 3 × 13
The LCM is the product of all the prime factors with their highest power:
LCM = 1 × 179 × 2 × 3 × 3 × 13 = 23814
Step 2: Subtract 3 from the LCM.
The greatest number that satisfies both conditions is obtained by subtracting 3 from the LCM:
23814 - 3 = 23811
Therefore, the greatest number that leaves a remainder of 3 when divided by both 179 and 234 is 23811.
To find the greatest number that satisfies the given conditions, we need to find the least common multiple (LCM) of the divisors (179 and 234) and subtract 3 from it.
First, let's calculate the LCM of 179 and 234:
1. Prime factorize 179: 179 is a prime number, so its prime factorization is 179.
2. Prime factorize 234: 234 can be factorized as 2 * 3 * 3 * 13.
Next, we identify the common and unique prime factors between the two numbers:
- The common factor between 179 and 234 is the number 3.
To calculate the LCM, we take the highest power of each prime factor:
- The highest power of 2 is 1 (from 2 * 3 * 3 * 13).
- The highest power of 3 is 2 (from 2 * 3 * 3 * 13).
- The highest power of 13 is 1 (from 2 * 3 * 3 * 13).
Multiplying these powers together, we get: LCM = 2 * 3^2 * 13 = 234 * 3 = 702.
Now, subtracting 3 from the LCM gives us the greatest number that satisfies the given conditions:
Greatest number = 702 - 3 = 699.
Therefore, the greatest number that, when divided by 179 and 234, leaves a remainder of 3 in each case is 699.