Quarter Wit Quarter Wisdom: What is Your Favorite Number?

Quarter Wit, Quarter WisdomFans of The Big Bang Theory will remember Sheldon Cooper’s quote from an old episode on his favorite number:

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 “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
(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 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 (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 FacebookYouTubeGoogle+, 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!

GMAT Tip of the Week

Think Globally, Act Locally, Succeed Exponentially

(This is one of a series of GMAT tips that we offer on our blog.)

As mentioned previously in this space, the GMAT’s quantitative section is increasingly emphasizing problem solving skills over calculation abilities, and often does so in the form of “Number Properties” questions. The authors of the exam are also quite adept at recognizing “mathematical psychology”, and creating questions that increase an examinee’s anxiety by enough to make that process of problem solving a bit more difficult. One of the major themes that arises as a result is the use of exponents, which carry with them a number of properties extremely useful to the writers of the GMAT.

Exponents:

  • Inspire fear (or at least apprehension) in test takers
  • Lead to cumbersome, time-consuming calculations involving large numbers
  • Are actually quite pattern-driven, and reward those who seek out those patterns rather than attempt to perform the extensive calculations

How can this help you on the exam?

If you embrace the pattern-driven quality of exponents, you can rest easy on exponent questions involving large numbers, knowing that you can test the pattern with small numbers, and simply extrapolate it to solve the overall question. Take, for example, a question that asks:

What is the sum of the digits of 10^25 – 37?

Listing out the numbers will be time consuming and contains the potential for error (counting to 25 when writing out the zeroes, then writing out that many digits in the difference, is a tedious process). But if you recognize that the first number will simply be “1” followed by 25 zeroes, and that the difference of the two will end in 63, preceded by a series of 9s, you can make quick work of this problem.

Try using a smaller exponent to see what the result will look like:

10^4 – 37:

10000
– 37
9963

In this case, if the exponent is 4, we end up with 4 digits in the answer: a 6, a 3, and the rest are 9s. Trying again with another exponent, we can see that the pattern holds for any exponent:

10^6 – 37:

1000000
– 37
999963

Again, we have the same number of digits in the difference as the value of the exponent, and two of those are 6 and 3, with the others 9s. So, we can conclude that 10^x – 37 will give us a solution in which we have (x-2) 9s, a 6, and a 3. Because 6+3 is 9, the sum of the digits will be (x-2) * 9 + 9, or (x-1) * 9.

In the original example, the exponent is 25, so we’ll have (25-1) * 9, or 24*9, and the answer is 216.

For more GMAT prep tips and resources, be sure to visit Veritas Prep. And, while you’re at it, follow us on Twitter!