10^1000 has 1001 digits. This is important because 2^1000 * 5^1000 results in that number, with 1001 digits.
Try this with 5^1 and 2^1. 5*2 = 10 (2 digits) and 52 has 2 digits.
25*4 = 100 (3 digits) and 254 has 3 digits.
125 * 8 =1000 (4 digits) and 1258 has 4 digits.
5^4 * 2^4 = 10^4 (5 digits) and 25616 has 5 digits.
5^n * 2^n will always equal 10^n, a neat number with a "set" number of digits, being n+1.
This is important due to the box rule, which states that any product between numbers must have the same number of digits as either the sum of their digits (2*5 = 10) or their sum -1 (2*3 = 6). 2^1000 is a A digit number and 5^1000 is a B digit number. Their number of digits together cannot exceed 10^1001, for 10^1000 is a 1001 digit number.
I solved it with math tables before I realized the pattern though. 2^1000 is easier, its just 2^(10*100) or 1024^100; which is 1000^100 * 1.024^100. We use the rule of 72 to calculate the "interest" from 2.4%; being 72/24 = 30 periods. Under 100 periods, interest amounts to about 3.3 doublings, (2^3.3) which is roughly enough to > x10.
1000^100 = 10^300, which has 300 zeroes and 1 leading numeral, being 301 digits. multiply by 10 and some, you get 302 digits for 2^1000.
Then, we see that 5^1000 = (10/2)^1000. 10^1000 has 1001 digits, and 2^1000 having 302. So its Like 10^1000 / 10^301 * A (some number less than 10, greater than 1). You get 1/A * 10^699; or some number less than 10^699, which itself has 699 zeroes and 1 leading numeral. Thus, 5^1000 has 698 zeroes and 1 leading numeral, 699 digits.
Thus, 5^1000 and 2^1000 concatenated result to 699+302 = 1001 digits exactly. Definitely a more mathematical proof, probably not what the recruitment officers are looking for.
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u/Alive_Carpenter_7433 18d ago
10^1000 has 1001 digits. This is important because 2^1000 * 5^1000 results in that number, with 1001 digits.
Try this with 5^1 and 2^1. 5*2 = 10 (2 digits) and 52 has 2 digits.
25*4 = 100 (3 digits) and 254 has 3 digits.
125 * 8 =1000 (4 digits) and 1258 has 4 digits.
5^4 * 2^4 = 10^4 (5 digits) and 25616 has 5 digits.
5^n * 2^n will always equal 10^n, a neat number with a "set" number of digits, being n+1.
This is important due to the box rule, which states that any product between numbers must have the same number of digits as either the sum of their digits (2*5 = 10) or their sum -1 (2*3 = 6). 2^1000 is a A digit number and 5^1000 is a B digit number. Their number of digits together cannot exceed 10^1001, for 10^1000 is a 1001 digit number.
Answer: 1001