If Two Positive Integers a and b are Chosen at Random between 1 and 50 GMAT Problem Solving

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.

Suggested GMAT Problem Solving Samples

• Three Cars Leave From A To B In Equal Time Intervals GMAT Problem Solving
• An Inlet Pipe can Fill in an Empty Cistern in 30 minutes GMAT Problem Solving
• A Zookeeper Counted the Heads of the Animals in a Zoo and Found it to be 80 GMAT Problem Solving
• A Shop Stores x kg of Rice. The First Customer Buys half this Amount Plus half a kg of Rice GMAT Problem Solving
• In a Class of 120 Students Numbered 1 to 120, All Even Numbered Students Opt for Physics GMAT Problem Solving
• Machine A Produces bolts at a Uniform Rate of 120 Every 40 seconds GMAT Problem Solving
• Out of 7 Consonants and 4 Vowels, How Many Words of 3 Consonants and 2 Vowels Can be Formed? GMAT Problem Solving
• 4 Bells Toll Together at 9:00 A.M. They Toll After 7, 8, 11 and 12 Seconds GMAT Problem Solving
• 12 Marbles are Selected at Random from a Large Collection of White, Red, Green and Yellow Marbles GMAT Problem Solving
• Find the greatest number that will divide 43, 91 and 183 GMAT Problem Solving
• Of the 150 Houses in a Certain Development GMAT Problem Solving
• A man can hit a target once in 4 shots. If he fires 4 shots in succession GMAT Problem Solving
• If 75 Percent of a Class Answered the First Question on a Certain Test Correctly GMAT Problem Solving
• A Clock Strikes 4 taking 9 seconds. GMAT Problem Solving
• What is the Largest Power of 3 Contained in 200! GMAT Problem Solving
• The square of 5√252 =? GMAT Problem Solving 
• GMAT Problem Solving - Which of the following expressions has the greatest values?
• GMAT Problem Solving – If @ x=x^2/2x^2-2 , What is the Units Digit of @ (@4)?
• GMAT Problem Solving – What is the product of all possible solutions of the equation |x+2|- 5|x+2| = -6?
• GMAT Problem Solving – Metropolis Corporation has 4 Shareholders
• GMAT Problem Solving – What is the number of integers from 1 to 1000, inclusive that are not divisible by 11 or by 35?
• GMAT Problem Solving – If m is Three Times n, and if 2n + 3 is 20% of 25, What is the value of m?
• GMAT Problem Solving – If Ben Were to Lose the Championship, Mike would be the Winner
• GMAT Problem Solving – A Train Travelling at a Certain Constant Speed takes 30 seconds
• GMAT Problem Solving – A conference room is equipped with a total of 45 metal or wooden chairs.
• A contractor estimated that his 10-man crew could complete the construction GMAT Problem Solving
• A circle is inscribed in a square with the diagonal of 4 centimeters GMAT Problem Solving
• At a dog competition, a dog is awarded 10 points if it runs through 4 pipes GMAT Problem Solving
• Two consultants, Mary and Jim, can type up a report in 12.5 hours and edit it in 7.5 hours GMAT Problem Solving
• Find The Value Of x GMAT Problem Solving
• 4 Out Of 15 Apples Are Rotten GMAT Problem Solving
• Last Year 26 Members of a Certain Club Traveled to England GMAT Problem Solving
• Running at the Same Constant Rate, 6 Identical Machines GMAT Problem Solving
• A conference room is equipped with a total of 45 metal or wooden chairs GMAT Problem Solving
• A welder received an order to make a 1 million litre cube-shaped tank GMAT Problem Solving
• After 6 games, Team B had an average of 61.5 points per game GMAT Problem Solving
• If 12 ounces of a strong vinegar solution are diluted with 50 ounces of water GMAT Problem Solving
• The square of \(5^\sqrt2)\ =? GMAT Problem Solving
• If 4 people are selected from a group of 6 married couples GMAT Problem Solving
• If the equation |x|+|y|= 5 encloses a certain region on the graph GMAT Problem Solving
• GMAT Problem Solving – A welder received an order to make a 1 million litre cube-shaped tank
• After 6 games, Team B had an average of 61.5 points per game GMAT Problem Solving 
• Simplify:\frac{4.5-2*\frac{3}{6}+\frac{1}{4^2}}{0.75} GMAT Problem Solving 
• There are 6 points on xy-plane GMAT Problem Solving 
• Mike, Tom, and Walt are working as sales agents for an insurance company GMAT Problem Solving 
• What is the value of k if the sum of consecutive odd integers from 1 to k equals 441? GMAT Problem Solving 
• A circle with a radius R is inscribed into a square with a side K GMAT Problem Solving 
• The square of \(5^\sqrt2)\ =? GMAT Problem Solving 
• A farm has chickens, cows and sheep GMAT Problem Solving 
• If g is an integer what is the value of (-1)^g^4-1? GMAT Problem Solving