when the integer n is divided by 8, the remainder is 3. what is the remainder if 6n is divided by 8?

The remainder is 2.

Quick Scoop: What’s Going On?

You’re told:

• When the integer nnn is divided by 8, the remainder is 3.
  That means nnn can be written as

n=8k+3n=8k+3n=8k+3

for some integer kkk.

You’re asked:

• What is the remainder when 6n6n6n is divided by 8?

Step-by-step Reasoning

1. Start with the expression for nnn:

n=8k+3n=8k+3n=8k+3

2. Multiply both sides by 6 to get 6n6n6n:

6n=6(8k+3)=48k+186n=6(8k+3)=48k+186n=6(8k+3)=48k+18

3. Now look at 48k+1848k+1848k+18 modulo 8:

   • 48k48k48k is clearly divisible by 8, since 48=8×648=8\times 648=8×6.
     So 48k48k48k leaves remainder 0 when divided by 8.

   • Focus on 18:

18=16+2=8×2+218=16+2=8\times 2+218=16+2=8×2+2

so 18 leaves remainder 2 when divided by 8.

4. Put it together:
   • 48k48k48k contributes remainder 0,
   • 18 contributes remainder 2,
     so 6n=48k+186n=48k+186n=48k+18 leaves remainder 2 when divided by 8.

Therefore, the remainder is 2.

Tiny Story to Remember It

Think of numbers that give remainder 3 when divided by 8:
3, 11, 19, 27, … Pick one, say n=11n=11n=11:

• 6n=6×11=666n=6\times 11=666n=6×11=66
• Divide 66 by 8:
  8×8=648\times 8=648×8=64, remainder 66−64=266-64=266−64=2.

Same remainder: 2. ✅ Answer: The remainder when 6n6n6n is divided by 8 is 2.