Quarter Wit Quarter Wisdom What is Your Favorite Number

Fans of The Big Bang Theory will remember Sheldon Cooper’s quote  from an old episode  on his favorite number: â€Å"The best number is 73. Why? 73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3 and in binary 73 is a palindrome, 1001001, which backwards is 1001001. Though Sheldon’s logic is infallible, my favorite number is 1001 because it has a special role in standardized tests. 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 GMAT questions and should be remembered as a â€Å"special† number. Lets 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 (E) 924 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! Let me explain: 1001 = 7 * 11 * 13  (Try dividing 1001 by 7 and youll 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 (discussed here  and continued  here). 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 (E) 924 Divisible by 7 Options B, C and E 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. I 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! Getting ready to take the GMAT? We have  free online GMAT seminars  running all the time. And, be sure to follow us on  Facebook,  YouTube,  Google+, and  Twitter! Karishma, a Computer Engineer with a keen interest in alternative Mathematical approaches, has mentored students in the continents of Asia, Europe and North America. She teaches the  GMAT  for Veritas Prep and regularly participates in content development projects such as this blog!