1001-Special Number

1001 is 1 more than 1000 and hence, is sometimes split as (1000 + 1). It sometimes appears in the a^2 – b^2 format such as 1001^2 – 1, and its factors are 7, 11 and 13 (not the factors we usually work with). Due to its unusual factors and its convenient location (right next to 1000), it could be a part of some tough-looking  questions and should be remembered as a “special” number. Let’s look at a question to understand how to work with this  number. Which of the following is a factor of 1001^(32) – 1 ? (A) 768 (B) 819 (C) 826 (D) 858 Note that 1001 is raised to the power 32. This is not an exponent we can easily handle. If  we try to use a binomial here and split 1001 into (1000 + 1), all we will achieve is that upon expanding the given expression, 1 will be cancelled out by -1 and all other terms will have 1000 in common. None of the answer choices are factors of 1000, however, so we must look for some other factor of 1001^(32) – 1. Without a calculator, it is not possible for us to find the factors of 1001^(32) – 1, but we do know the prime factors of 1001 and hence, the prime factors of 1001^32. We may not be able to say which numbers are factors of 1001^(32) – 1, but we will be able to say which numbers are certainly not factors of this number. Explaination: 1001 = 7 * 11 * 13 (Try dividing 1001 by 7 and you’ll get 143. 143 is divisible by 11, giving you 13.) 1001^32 = 7^32 * 11^32 * 13^32 Now, what can we say about the prime factors of 1001^(32) – 1? Whatever they are, they are certainly not 7, 11 or 13 – two consecutive integers cannot have any common prime factor. Now look at the answer choices and try dividing each by 7: (A) 768 – Not divisible by 7 (B) 819 – Divisible by 7 (C) 826 – Divisible by 7 (D) 858 – Not divisible by 7 Options B and C are eliminated. They certainly cannot be factors of 1001^(32) – 1 since they have 7 as a prime factor, and we know 1001^(32) – 1 cannot have 7 as a prime factor. Now try dividing the remaining options by 11: (A) 768 – Not divisible by 11 (D) 858 – Divisible by 11 D can also be eliminated now because it has 11 as a factor. By process of elimination, the answer is A; it must be a factor of 1001^(32) – 1. Hope you see how easily we used the factors of 1001 to help us solve this difficult-looking question. And yes, another attractive feature of 1001 – it is a palindrome in the decimal representation itself!