Question: If two positive integers a and b are chosen at random between 1 and 50 inclusive, what is the approximate probability that a number of the form \(7^a+7^b\) is divisible by 5?

1. \(\frac{1}{5}\)
2. \(\frac{1}{4}\)
3. \(\frac{1}{2}\)
4. \(\frac{2}{3}\)
5. \(\frac{3}{4}\)

Answer:
Solution with Explanation:
Approach Solution (1):

\(7^1\)mod 5 = 2

\(7^2\)mod 5 = -1

\(7^3\)mod 5 = -2

\(7^4\)mod 5 = 1

\(7^5\)mod 5 = 2; cyclicity of 4

Combining \(7^1\) and \(7^3\) we will leave a zero remainder when divided by 5. Similarly, combining \(7^2\)and \(7^4\) will leave no remainder when divided by 5.

So, we need to combine powers of the form 4k+1 and 4k+3 or 4k+2 and 4k+4

Between 1 and 50 (inclusive), there will be 13 numbers of the form 4k+2 and 12 numbers of the form 4k+4

So the total number of ways for selecting the exponents for a zero remainder – 13 * 12 + 13 * 12 = 13 *24

The total number of ways for selecting two integers from 1-50 (inclusive) = \(^{50}C_2 \)= 25 * 49

Probability = \(13* \frac{24}{25}*49 \approx \frac{1}{4}\)

Correct Option: B

Approach Solution (2):

There are 4 possibilities when the sum (given in the question) will be divisible by 5, combinations of 7 and 3, 9 and 1, 3 and 7 and 1 and 9.

Again the periodicity of the repetition of power of 7 is 4, i.e., every 1st, 5th, 9th… (and so on) time the unit digit will be 7, 2nd, 6th time… will be ‘9’, 3rd, 7th time… will be ‘3’ and likewise.

So the probability to get each of the unit digit is 12 ( \(\frac{50}{4}\)approx).

Probability for \(7^a\)= \(\frac{12}{50}\)

Similarly for \(7^b\)= \(\frac{12}{50}\)

Also, there are a total of 4 combinations. Hence the combined probability becomes\(\frac{12}{50}\)*\(\frac{12}{50}\)*4 = \(\frac{1}{4}\)*\(\frac{1}{4}\)*4=\(\frac{1}{4}\)

Correct Option: B

“If two positive integers a and b are chosen at random between 1 and 50 inclusive, what is the approximate probability that a number of the form \(7^a+7^b\) is divisible by 5??”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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