Finding the Least Common Multiple (LCM) using Remainders

To find the least multiple of 7 that leaves a remainder of 4 when divided by 6, 9, 15, and 18, we need to find the LCM of these four numbers and then add 4 to it. However, instead of finding the LCM directly, we can use the remainders to simplify the problem.

Step 1: Find Numbers Divisible by All Four Divisors

Let x be the least multiple of 7 that leaves a remainder of 4 when divided by 6, 9, 15, and 18. Then we can write:

x ≡ 4 (mod 6)
x ≡ 4 (mod 9)
x ≡ 4 (mod 15)
x ≡ 4 (mod 18)

Since 6, 9, 15, and 18 are all divisors of some number, we can find numbers that are divisible by all four of them. In particular:

LCM(6, 9, 15, 18) = 270

So, we can write:

x ≡ 4 (mod 270)

Step 2: Find Smaller Modulus

Now we need to find a smaller modulus that satisfies the same congruences. To do this, we can find the GCD (greatest common divisor) of the differences between the moduli:

gcd(6, 9, 15, 18) = 3

So, we divide the modulus by 3 to get:

270/3 = 90

So, we can write:

x ≡ 4 (mod 90)

Step 3: Find Least Multiple

Finally, we find the least multiple of 7 that leaves a remainder of 4 when divided by 90. We can do this by listing multiples of 7 until we find one that leaves a remainder of 4 when divided by 90. Alternatively, we can use the fact that 7 and 90 are coprime (i.e., they have no common factors other than 1) to find the inverse of 7 modulo 90:

7x ≡ 1 (mod 90)

Solving this congruence using the Extended Euclidean Algorithm, we get:

x ≡ 13 (mod 90)

So, the least multiple of 7 that leaves a remainder of 4 when divided by 6, 9, 15, and 18 is:

x = 13 + 90n

where n is an integer. The smallest such multiple greater than 0 is:

x = 13 + 90 = 103

Therefore, the correct option is (d) 364.