To determine the sum of all single-digit replacements for n such that the number {42,789,n37} is divisible by 3, you need to find the values of n that make the sum of the digits divisible by 3. Here are the steps:
First, find the sum of the digits in the number {42,789,n37} without replacing n.
4 + 2 + 7 + 8 + 9 + n + 3 + 7 = 30 + n
Now, you want the sum of the digits to be divisible by 3, so you need to find the possible values of n that make 30 + n divisible by 3.
30 + n ≡ 0 (mod 3)
To make 30 + n divisible by 3, n must be equal to -30 modulo 3.
n ≡ -30 (mod 3)
Find the remainder when -30 is divided by 3:
-30 ÷ 3 = -10
So, n ≡ -10 (mod 3).
Now, you need to find all single-digit replacements for n that satisfy this congruence:
n ≡ -10 (mod 3)
The single-digit numbers that satisfy this congruence are:
n = 1 (because -10 + 1 = -9, which is divisible by 3)
n = 4 (because -10 + 4 = -6, which is divisible by 3)
n = 7 (because -10 + 7 = -3, which is divisible by 3)
Now, calculate the sum of these possible values for n:
1 + 4 + 7 = 12
So, the sum of all single-digit replacements for n such that the number {42,789,n37} is divisible by 3 is 12.