7 d ≡ 1 mod 40

To solve this congruence, we need to find the value of d that satisfies the equation 7 * d ≡ 1 (mod 40).

First, let's find the modular inverse of 7 modulo 40. The modular inverse of a number a modulo m is the number b such that (a * b) ≡ 1 (mod m).

In this case, we need to find the modular inverse of 7 modulo 40. We can use the Extended Euclidean Algorithm to find the modular inverse.

40 = 7 * 5 + 5 7 = 5 * 1 + 2 5 = 2 * 2 + 1

Now, we can work our way back up the algorithm to find the modular inverse:

1 = 5 - 2 * 2 = 5 - (7 - 5 * 1) * 2 = 5 * 3 - 7 * 2

Therefore, the modular inverse of 7 modulo 40 is 3.

Now, we can multiply both sides of the congruence by the modular inverse to isolate d:

7 * d * 3 ≡ 1 * 3 (mod 40) 21 * d ≡ 3 (mod 40)

To find the value of d, we can multiply both sides of the congruence by the modular inverse of 21 modulo 40, which we can find using the Extended Euclidean Algorithm:

40 = 21 * 1 + 19 21 = 19 * 1 + 2 19 = 2 * 9 + 1

1 = 19 - 2 * 9 = 19 - (21 - 19 * 1) * 9 = 19 * 10 - 21 * 9

Therefore, the modular inverse of 21 modulo 40 is 10.

Multiplying both sides of the congruence by 10:

21 * d * 10 ≡ 3 * 10 (mod 40) 210 * d ≡ 30 (mod 40)

Reducing the equation:

10 * d ≡ 30 (mod 40)

Now, we can solve for d by dividing both sides of the congruence by 10:

d ≡ 3 (mod 40)

Therefore, the value of d that satisfies the congruence 7 * d ≡ 1 (mod 40) is d ≡ 3 (mod 40)