Question

Find the unit digit of the largest prime number known:
2^(13466917) – 1
Hint : This number contains 4,053,946 digits!

Answer 1

Question

Find the unit digit of $2^{13466917}-1$.

Keys

• Base 10

It is one of the number systems.

The place values for 1 in each digit is a power of 10.

Ex. $1111=1\cdot10^3+1\cdot10^2+1\cdot10^1+1\cdot10^0$

• Cycle

If we know the period of the cycle, we can specify a number.

Solution

(Step I) Division by 10

Let's multiply the first digit and a higher digit. The product ends at least in the tens digit.

Then, we ignore the higher digit and only multiply the unit digit.

$2$ → $4$ → $8$ → $16$(or $6$) → $12$(or $2$)

The pattern is 2, 4, 8, 6.

(Step II) Cycle

The unit digit cycles every 4th number.

To find which between 2, 4, 8, 6 is the unit digit of $2^n$, we divide $n$ by 4.

We divide the last two digits by 4.[1] The remainder is 1.

The unit digit of $2^{13466917}$ is 2.

Conclusion

$2^{13466917}-1$ has the unit digit of 1.

More information

The number of digits in $2^{13466917}-1$

What is the number of the digits of the 39th Mersenne prime?[2]

$\log2^{13466917}$

$=13466917 \cdot \log2$

$\approx 4053945$ (Rounded down to the unit digit)

Since the unit digit is not 0, there won't be a carry-over. We conclude the  number of the digits is $4053946$.

[1] 100 is the first power of 10 that is divisible by 4. The remainder comes from the last two digits.

[2] Since we use base 10, we need the highest power to find the number of the digits. Power can be found by logarithm. Then 1 is added to the power.

Answer 2

Question:-

• Find the unit digit of the largest prime number known:
• 2^(13466917) – 1

To Find:-

• Find the largest number.

Solution:-

Here ,

There are 2 steps . They are:-

Step 1 : - First , we have to multiply first with the higher digit . Then the product end at least in tens digit.

So ,

2 → 4 → 8 → 16( or 6 ) → 12( or 2 )

The pattern is 2 , 4 , 8 , 6 .

Step 2 : - The unit digit cycles every 4th number.

Now ,

We need to find unit number so to find which is between 2 , 4 , 8 , 6 is the unit digit of 2^n , and then we have to divide by n / 4 .

Hence ,

• $\tt \: The unit digit is 1 .$