Question: What is the unit’s digit of \((17^3)^4-{1973^3}^2\) ?
- 3
- 7
- 15
- 21
- 25
Answer:
Approach Solution 1:
We know that:
(1) The unit’s digit of \((abc)^n\)is the same way as that of \(c^n\) which means that the unit’s digit of \((17^3)^4\) is the same as that of \((17^3)^4\) and the units digit of \({1973^3}^2\) is the same as that of \({3^3}^2\).
(2) If exponentiation is indicated by stacked symbols, the rule is to work from top to down, thus: \({a^m}^n=a({^m}^n)\) and not \({a(^m)}^n\) , which on the other hand equals to \(a(^{mn})\)
So:
\((7^3)^4=(7)^{3*4}=7^{12}and3^{3^2}=3(^{3^2})=3^9\)
(3) The unit’s digit of the integers in positive integer power repeats in specific pattern: The unit digit of 7 and 3 in positive integer power repeats in patterns of 4:
- \(7^1=7\)(Last digit is 7)
- \(7^2=49\)(Last digit is 9)
- \(7^3=xx3\)(Last digit is 3)
- \(7^4=xxx1\)(Last digit is 1)
- \(7^5=xxxxxx7\)(Last digit is 7 again)
…
- \(3^1=7\)(Last digit is 3)
- \(3^2=9\)(Last digit is 9)
- \(3^3=27\)(Last digit is 7)
- \(3^4=81\)(Last digit is 1)
- \(3^5=243\)(Last digit is 3 again)
…
Thus, the unit’s digit of \(7^{12}\) will be 1 (4th in pattern, as 12 is a multiple of cyclist number 4) and the unit’s digit of \(3^9\) will be 3 (first in pattern, as 9 = 4 * 2 + 1)
So, we have that the unit’s digit of \((17^3)^4\) = \((17)^{12}\) is 1 and the unit’s digit of \({1973^3}^2={1973^9}\) is 3. Also notice that the second number is much larger than the first one, thus their difference will be negative, something like 11 – 13 = -2, which gives the final answer that the unit’s digit of \((17^3)^4-{1973^3}^2\)is 2.
Correct option: B
“What is the unit’s digit of?”- 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 Quant Questions:
- A rectangle is inscribed in a hexagon that has all sides of equal length and all angles of equal measure GMAT Problem Solving
- If x = -3, What Is The Value Of -3x^2? GMAT Problem Solving
- There Were R Red Balls And Y Yellow Balls In A Bag. Three Red Balls GMAT Problem Solving
- With An Average Speed Of 40 Km/H, A Train Reaches Its Destination On GMAT Problem Solving
- A Photographer Will Arrange 6 People Of 6 Different Heights GMAT Problem Solving
- What Is The Radius Of The Incircle Of The Triangle Whose Sides Measure GMAT Problem Solving
- The Value Of (2^(-14) + 2^(-15) + 2^(-16) + 2^(-17))/5 Is GMAT Problem Solving
- Points A And B Are 120 Km Apart. A Motorcyclist Starts From GMAT Problem Solving
- A student took five papers in an examination, where the full marks GMAT Problem Solving
- In how many ways can letters the word ATTITUDE be rearranged such that GMAT Problem Solving
- A merchant mixes three varieties of rice costing $20/kg, $24/kg GMAT Problem Solving
- ABC is an equilateral triangle, and point D is the midpoint of side BC GMAT Problem Solving
- A Batsman Makes a Score of 87 Runs in the 17th Match and Thus Increases GMAT Problem Solving
- If M= √4+3√4+4√4, Then the Value of M is GMAT Problem Solving
- An Octagon Is Inscribed In A Circle As Shown Above. What Of The Area GMAT Problem Solving
- In a Company of Only 20 Employees, 10 Employees make $80,000/yr GMAT Problem Solving
- A bag contains blue and red balls only GMAT Problem Solving
- (4.8*10^9)^(1/2) is closest in value to GMAT Problem Solving
- What Is The Units Digit Of 2222^333 ∗ 3333^222? GMAT Problem Solving
- What Is The Tens Digit Of 6^17? GMAT Problem Solving
- If m=−2, What Is −m^(−m)? GMAT Problem Solving
- How many Terminating Zeroes does 200 Have GMAT Problem Solving
- Properties of Circle GMAT Problem Solving
- If 10, 12 and ‘x’ are Sides of an Acute Angled Triangle, How Many Integer Values of ‘x’ are Possible? GMAT Problem Solving
- For How Many Values of k is 12^12 the Least Common Multiple GMAT Problem Solving
- Bag A Contains Red, White and Blue Marbles such that GMAT Problem Solving
- Assume that all 7-Digit Numbers That do not Begin with 0 or 1 are Valid Phone Numbers. GMAT Problem Solving
- A Car Travels from Mayville to Rome at an Average Speed of 30 miles per hour GMAT Problem Solving
- A Certain Sum of Money is Divided Among A, B and C such that A Gets One GMAT Problem Solving
- The Ratio of Boys to Girls in Class A is 1 to 4, and that in Class B is 2 to 5 GMAT Problem Solving
- The Maximum Mark in an Examination is 100 and the Minimum is 0 GMAT Problem Solving
- A Rectangular Box has Dimensions 12*10*8 Inches GMAT Problem Solving
- A Driver Completed the First 20 Miles of a 40-Mile Trip at an Average Speed of 50 Miles Per Hour GMAT Problem Solving
- The sum of three numbers is 98. If the ratio between first and second be 2:3 and between second and third be 5:8 GMAT Problem Solving
- How Many Three-Letter Words Can be Constructed Using All the 26 Letters of the English Alphabet GMAT Problem Solving
- How Many Litres of Pure Alcohol Must be Added to a 100-litre Solution That is 20 Percent Alcohol GMAT Problem Solving
- For Any Four Digit Number, abcd, *abcd*= (3^a)(5^b)(7^c)(11^d) GMAT Problem Solving
- How Many Five Digit Numbers Can be Formed Using Digits 0, 1, 2, 3, 4, 5, Which Are Divisible By 3 GMAT Problem Solving
- An “Armstrong Number” is an n-Digit Number That is Equal to the Sum of the nth Powers GMAT Problem Solving
- A train crosses a bridge of length 500 m in 40 seconds and a lamp post on the bridge in 15 seconds
- A Train can Travel 50% Faster than a Car GMAT Problem Solving
- An Automated Manufacturing Plant Uses Robots To Manufacture Products GMAT Problem Solving
- The Surface Distance Between 2 Points on the Surface of a Cube is the GMAT Problem Solving
- The Average Monthly Expenditure of a Family for the First Four Months GMAT Problem Solving
- When a Certain Perfect Square is Increased by 148, the Result is GMAT Problem Solving
- If p#q Denotes the Least Common Multiple of p and q, Then ((12#16) GMAT Problem Solving
- The Smallest of Six Consecutive Odd Integers Whose Average (arithmetic mean) is x + 2 GMAT Problem Solving
- The Greatest 6-Digit Number When Divided by 6, 7 ,8 , 9, and 10 Leaves a Remainder of 4, 5, 6, 7, and 8 Respectively GMAT Problem Solving
- Is Zero Even Integer or Odd Integer? GMAT Problem Solving
- If 20 Men or 24 Women or 40 Boys can do a Job in 12 Days GMAT Problem Solving
*The article might have information for the previous academic years, please refer the official website of the exam.