Okay, so I need to convert 555 from base 10 to base 5. Hmm, let me remember how to do this. I think the standard method involves dividing the number by 5 repeatedly and keeping track of the remainders. Let me try to recall the steps.

First, you divide 555 by 5. The quotient is the next number to divide, and the remainder becomes a digit in the base 5 number. Then you repeat this process until the quotient is zero. Finally, you write the remainders in reverse order to get the base 5 number.

Let me start with 555 divided by 5. 555 divided by 5 is 111 with a remainder of 0. So the first remainder is 0. Then take 111 and divide by 5. 111 divided by 5 is 22 with a remainder of 1. Next, take 22 divided by 5. 22 divided by 5 is 4 with a remainder of 2. Then take 4 divided by 5. 4 divided by 5 is 0 with a remainder of 4. Since the quotient is now 0, we stop.

So the remainders we got were 0, 1, 2, 4. But since we write them in reverse order, the base 5 number should be 4 2 1 0. Wait, let me check that. Let me write them down in the order they were obtained: first remainder is the least significant digit. So the first remainder is 0 (units place), then 1 (5s place), then 2 (25s place), then 4 (125s place). So putting them together, it's 4*125 + 2*25 + 1*5 + 0*1. Let me calculate that: 4*125 is 500, 2*25 is 50, 1*5 is 5, and 0*1 is 0. Adding those up: 500 + 50 is 550, plus 5 is 555. That checks out. So 555 in base 5 is 4210 base 5.

Wait, but let me verify this another way. Maybe using another method to ensure I didn't make a mistake. Another way to convert is to express 555 as a sum of powers of 5. Let's see. The powers of 5 are 1, 5, 25, 125, 625, etc. Since 555 is less than 625, the highest power we need is 125 (5^3). Let's see how many times 125 fits into 555. 555 divided by 125 is 4 with a remainder of 555 - 4*125 = 555 - 500 = 55. Then 55 divided by 25 (5^2) is 2 with a remainder of 55 - 2*25 = 55 - 50 = 5. Then 5 divided by 5 (5^1) is 1 with a remainder of 0. So that gives us coefficients 4, 2, 1, 0 for the powers 125, 25, 5, 1. So that's 4*125 + 2*25 + 1*5 + 0*1 = 500 + 50 + 5 + 0 = 555. So that's the same result as before. So that seems correct.

Alternatively, maybe I can check using another approach, like converting to binary first and then to base 5, but that might be more complicated. Let me try that. 555 in binary. Let's see. 512 is 2^9, which is 512. 555 - 512 = 43. 32 is 2^5, so 43 -32=11. 8 is 2^3, 11-8=3. 2 is 2^1, 3-2=1. 1 is 2^0. So binary is 100010011. Let me confirm: 512 + 32 + 8 + 2 + 1 = 512+32=544, 544+8=552, 552+2=554, 554+1=555. Yes. So binary is 100010011. Now, to convert binary to base 5. Since 5 is 2^2 + 1, maybe we can group bits. Each base 5 digit corresponds to 2 binary digits. Let me check. Let's split the binary number into groups of 2 from the right. 100010011. Starting from the right: 11, 01, 00, 10, 01. Wait, but 100010011 has 9 bits. So grouping into 2s from the right: 11, 01, 00, 10, 01. But wait, the leftmost group is only 1 bit. Hmm, maybe I need to pad with a leading zero. So adding a leading zero gives 0100010011. Then grouping into 2s: 01, 00, 01, 00, 11. Now each pair is 01, 00, 01, 00, 11. Convert each pair to base 5. Let's see: 01 is 1, 00 is 0, 01 is 1, 00 is 0, 11 is 3. So the base 5 number is 101013. Wait, but that's 5 digits: 1, 0, 1, 0, 1, 3. Wait, wait, no. Wait, the binary number is 100010011, which is 9 bits. When grouping into 2s from the right, with padding, the groups are 01, 00, 01, 00, 11. Wait, no, 100010011 is 1 0001 0011. Wait, maybe my grouping was wrong. Let me write the binary number as 100010011. Let's count the bits: 1 (1), 0 (2), 0 (3), 0 (4), 1 (5), 0 (6), 0 (7), 1 (8), 1 (9). So from the right, group into 2s: bits 9-8: 1,1; 7-6: 0,0; 5-4:1,0; 3-2:0,0; 1:1. Wait, that's not even. Hmm, maybe I need to split differently. Alternatively, maybe it's easier to convert binary to decimal first and then to base 5. But since we already have the base 5 result from the first method, maybe this is overcomplicating. But let's check. The binary number is 100010011. Let's convert that to decimal. Let's compute the value. The bits are from left to right: 1 (2^8), 0 (2^7), 0 (2^6), 0 (2^5), 1 (2^4), 0 (2^3), 0 (2^2), 1 (2^1), 1 (2^0). So that's 256 + 0 + 0 + 0 + 16 + 0 + 0 + 2 + 1 = 256 + 16 = 272, 272 + 2 = 274, 274 +1=275. Wait, but 275 is not 555. Wait, that can't be. Wait, no, 555 is 512 + 32 + 8 + 2 + 1 = 555. Wait, 2^9 is 512, so 512 + 32 is 544, +8 is 552, +2 is 554, +1 is 555. So binary is 100010011, which is 9 bits. So converting that to decimal should be 555. But when I computed the bits from left to right as 2^8 + 2^4 + 2^1 + 2^0, that's 256 + 16 + 2 +1=275. Wait, that's not matching. Wait, maybe I messed up the bit positions. Let me write the binary number again: 100010011. Let's list the bits from right to left, starting at position 0. So the rightmost bit is position 0: 1. Then position 1: 1, position 2:0, position3:0, position4:1, position5:0, position6:0, position7:0, position8:1. So the value is 1*2^8 + 0*2^7 + 0*2^6 + 0*2^5 + 1*2^4 + 0*2^3 + 0*2^2 + 1*2^1 +1*2^0. So that's 256 + 0 +0 +0 +16 +0 +0 +2 +1= 256+16=272+2=274+1=275. Wait, but that's 275, not 555. That's a problem. So something's wrong here. Wait, but 100010011 in binary is 275, but 555 in base 10 is 555. So how come the binary conversion gave me 275? That means I must have made a mistake in converting 555 to binary. Let me check again. Let's divide 555 by 2, keeping track of the remainders. 555 divided by 2 is 277 with remainder 1. 277 divided by 2 is 138 rem 1. 138 divided by 2 is 69 rem 0. 69 divided by 2 is 34 rem 1. 34 divided by 2 is 17 rem 0. 17 divided by 2 is 8 rem 1. 8 divided by 2 is 4 rem 0. 4 divided by 2 is 2 rem 0. 2 divided by 2 is 1 rem 0. 1 divided by 2 is 0 rem 1. So the remainders, from last to first, are 1,0,0,0,1,0,1,1,0,1. Wait, let's list them in order: 1 (from last division), then 1, then 0, then 0, then 0, then 1, then 0, then 1, then 1, then 0. Wait, that's 10 divisions. Wait, the remainders are collected from last to first. Let me list them step by step:

555 /2 = 277 rem 1

277 /2 =138 rem 1

138 /2=69 rem 0

69 /2=34 rem 1

34 /2=17 rem 0

17 /2=8 rem 1

8 /2=4 rem 0

4 /2=2 rem 0

2 /2=1 rem 0

1 /2=0 rem 1

So the remainders, from first to last, are 1,1,0,1,0,1,0,0,0,1. Wait, but when you collect remainders, the first remainder is the least significant bit. So the binary number is the remainders read from last division to first. So the remainders in order are 1,0,0,0,1,0,1,1,0,1. Wait, let me see. The first division gives remainder 1 (LSB), then 1, then 0, then 1, then 0, then 1, then 0, then 0, then 0, then 1 (MSB). So the binary number is 100010011. Wait, but when I compute that, it's 275. That contradicts. Wait, but 555 is 555, but 100010011 is 275. So there must be a mistake here. Wait, maybe I messed up the remainders. Let me check again. Let's do the divisions step by step:

555 /2 = 277.5, but since we need integer division, it's 277 with remainder 1 (since 2*277=554, 555-554=1). So remainder 1.

277 /2 =138.5, so 138*2=276, remainder 1.

138 /2=69 rem 0.

69 /2=34 rem 1.

34 /2=17 rem 0.

17 /2=8 rem 1.

8 /2=4 rem 0.

4 /2=2 rem 0.

2 /2=1 rem 0.

1 /2=0 rem 1.

So the remainders, in order from first to last (LSB to MSB), are 1,1,0,1,0,1,0,0,0,1. So writing them as bits from MSB to LSB, it's 100010011. Wait, but that's 9 bits. Let me compute that:

1*2^8 =256

0*2^7=0

0*2^6=0

0*2^5=0

1*2^4=16

0*2^3=0

0*2^2=0

1*2^1=2

1*2^0=1

Adding up: 256 +16 +2 +1=275. So that's 275, but we started with 555. That's a problem. So where is the mistake here? Wait, maybe the initial conversion is wrong. Let me check 275 in decimal is 100010011 in binary, which is 275. So 555 must be incorrect. Wait, no, 555 in base 10 is 555. Wait, maybe I messed up the initial division steps. Let me try converting 555 again.

Wait, perhaps I confused the order of remainders. Let me try a different approach. Let's divide 555 by 2 repeatedly and collect the remainders:

555 ÷2 = 277, remainder 1

277 ÷2 =138, remainder 1

138 ÷2=69, remainder 0

69 ÷2=34, remainder 1

34 ÷2=17, remainder 0

17 ÷2=8, remainder 1

8 ÷2=4, remainder 0

4 ÷2=2, remainder 0

2 ÷2=1, remainder 0

1 ÷2=0, remainder 1

So the remainders, from first to last, are 1,1,0,1,0,1,0,0,0,1. Wait, but the binary number should be written from last remainder to first. So the last remainder is 1, then 0, 0, 0, 1, 0, 1, 0, 1, 1. Wait, that's 10 bits. Wait, let me list them in reverse order of the remainders collected. The first remainder is the first division, which is the least significant bit. So the binary number is built by taking the remainders from last division to first. So the remainders collected in order are:

1 (from 555/2)

1 (277/2)

0 (138/2)

1 (69/2)

0 (34/2)

1 (17/2)

0 (8/2)

0 (4/2)

0 (2/2)

1 (1/2)

So the remainders in the order they were obtained are: 1,1,0,1,0,1,0,0,0,1. So to get the binary number, we need to reverse this order. So the last remainder is 1 (from 1/2), then 0 (from 2/2), then 0 (4/2), then 0 (8/2), then 1 (17/2), then 0 (34/2), then 1 (69/2), then 0 (138/2), then 1 (277/2), then 1 (555/2). Wait, but that would make the binary number 1000100111, which is 10 bits. Wait, but 555 is less than 512*2=1024, so the binary should be 10 bits. Let me check 1000100111 in binary. Let's compute:

1*512 +0*256 +0*128 +0*64 +1*32 +0*16 +0*8 +1*4 +1*2 +1*1 =512 +32 +4 +2 +1=512+32=544+4=548+2=550+1=551. That's 551, not 555. So that's still not matching. Hmm, this is confusing. Wait, perhaps I need to check the division steps again. Let me try dividing 555 by 2 step by step:

555 ÷2 =277, rem 1

277 ÷2=138, rem 1

138 ÷2=69, rem 0

69 ÷2=34, rem 1

34 ÷2=17, rem 0

17 ÷2=8, rem 1

8 ÷2=4, rem 0

4 ÷2=2, rem 0

2 ÷2=1, rem 0

1 ÷2=0, rem 1

So the remainders are: 1,1,0,1,0,1,0,0,0,1. The binary number is these remainders read from last to first, so the last remainder is 1 (from 1/2), then 0 (2/2), 0 (4/2), 0 (8/2), 1 (17/2), 0 (34/2), 1 (69/2), 0 (138/2), 1 (277/2), 1 (555/2). So the binary number is 1 0 0 0 1 0 1 0 1 1. Wait, that's 10 bits: 1000101011. Let's compute that:

1*512 +0*256 +0*128 +0*64 +1*32 +0*16 +1*8 +0*4 +1*2 +1*1 =512 +32 +8 +2 +1=512+32=544+8=552+2=554+1=555. Yes! That's correct. So the binary representation is 1000101011. But earlier, when I tried to group them into 2s for base 5, I got 101013, which when converted gives 1*125 +0*25 +1*5 +3*1=125+5+3=133, which is wrong. So that approach is flawed. Therefore, the initial method of dividing by 5 is correct, giving 4210 base 5. So why did grouping into 2s fail? Because when converting to base 5, the grouping method works only if the base is a power of 2, but 5 is not a power of 2. Therefore, that method is incorrect. So the first method is the correct one. Therefore, the correct answer is 4210 in base 5. Therefore, the answer is $\boxed{4210_5}$.

**Final Answer**
\boxed{4210_5}
Therefore, the base $5$ representation of $555_{10}$ is $\boxed{4210_5}$.
</think>