Question

Find the greatest number which when divided by 179 and 234 leaves a remainder of 3 in each case

Answer 1

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.