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10 Questions MCQ Test Practice Questions for GMAT - Test: Exponents/Powers

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Test: Exponents/Powers - Question 1

What is the remainder when 337 is divided by 10 ?

Detailed Solution for Test: Exponents/Powers - Question 1

1. Understanding the Pattern of Last Digits

When dealing with powers of a number and division by 10, it's helpful to observe the pattern of the last digits of the number's powers. This is because dividing by 10 essentially gives you the last digit of the number.

Let's list the last digits of the first few powers of 3:

  • 3¹ = 3 → Last digit is 3
  • 3² = 9 → Last digit is 9
  • 3³ = 27 → Last digit is 7
  • 3⁴ = 81 → Last digit is 1
  • 3⁵ = 243 → Last digit is 3
  • 3⁶ = 729 → Last digit is 9
  • 3⁷ = 2187 → Last digit is 7
  • 3⁸ = 6561 → Last digit is 1

Observation: The last digits repeat in a cycle of four: 3, 9, 7, 1.

2. Determining the Position in the Cycle

Since the pattern of last digits repeats every four powers, we can determine where 337 falls within this cycle by dividing the exponent (37) by 4 and finding the remainder.

  • Divide 37 by 4:
    • 37 divided by 4 equals 9 with a remainder of 1.

This means 337 corresponds to the first position in our four-number cycle.

3. Finding the Remainder

From our observed pattern:

  • Remainder 1: Last digit is 3
  • Remainder 2: Last digit is 9
  • Remainder 3: Last digit is 7
  • Remainder 0: Last digit is 1 (since a remainder of 0 means it completed the cycle)

Since the remainder is 1, the last digit of 337 

4. Conclusion

The last digit of 337 is 3, which means when you divide 337 by 10, the remainder is 3.

Final Answer

The remainder when 337 is divided by 10 is 3.

Test: Exponents/Powers - Question 2

If 17!/7m is an integer, what is the greatest possible value of m ?

Detailed Solution for Test: Exponents/Powers - Question 2

To determine the greatest possible value of m in the expression 17!/7m, we need to find the highest power of 7 that divides evenly into 17!.

The prime factorization of 17! can be determined by breaking down each number from 1 to 17 into its prime factors and multiplying them together. However, instead of going through the entire prime factorization process, we can count the number of factors of 7 that appear in the prime factorization of each number from 1 to 17.

We have the following numbers in the range 1 to 17 that are divisible by 7: 7, 14.

Each occurrence of 7 in the prime factorization contributes one factor of 7 to the overall count. Therefore, we have 2 factors of 7 in the prime factorization of 17!.

To ensure that the expression 17!/7m is an integer, we want to divide 17! by the highest power of 7, which is 72. Therefore, m should be set to the exponent of the highest power of 7, which is 2.

Hence, the greatest possible value of m is 2.

The correct answer is A.

Test: Exponents/Powers - Question 3

How many digits are there in the product 218∗517 ?

Test: Exponents/Powers - Question 4

What is the largest power of 5! that can divide 41!?

Detailed Solution for Test: Exponents/Powers - Question 4

To find the largest power of 5! that can divide 41!, we need to determine the number of times 5! (which is equal to 5 * 4 * 3 * 2 * 1 = 120) can be divided evenly into 41!.

Let's calculate the power of 5 in the prime factorization of 41!. We know that each multiple of 5 contributes at least one power of 5, and multiples of 25 contribute an additional power of 5.

To calculate the power of 5, we divide 41 by 5 and take the floor value:

41 ÷ 5 = 8

This means there are at least 8 powers of 5 in the prime factorization of 41!.

To account for the multiples of 25, we divide 41 by 25 and take the floor value:

41 ÷ 25 = 1

This means there is an additional power of 5 from the multiples of 25.

Therefore, the largest power of 5! that can divide 41! is 8 + 1 = 9.

The correct answer is A.

Test: Exponents/Powers - Question 5

What is the greatest prime factor of 417−228 ?

Detailed Solution for Test: Exponents/Powers - Question 5

4^17 can be written as 2^34, which gives us:
2^34 - 2^28
I can factor out a 2^28 to get:
2^28(2^6-1)
which is basically a bunch of 2's multiplied by (2^6 - 1)
The first part (the bunch of 2's) has 2 as the only prime factor.
The second part (2^6 - 1) = 64 -1 = 63. 63's prime factorization is 3*3*7. So 7 is the largest prime factor.

Test: Exponents/Powers - Question 6

What is the decimal equivalent of (2/5)5 ?

Detailed Solution for Test: Exponents/Powers - Question 6

To find the decimal equivalent of (2/5)5, we need to raise 2/5 to the power of 5 and evaluate the result.

(2/5)5 = (25)/(55) = 32/3125

Dividing 32 by 3125, we get:

32 ÷ 3125 = 0.01024

Therefore, the decimal equivalent of (2/5)5 is 0.01024.

The correct answer is E.

Test: Exponents/Powers - Question 7

The population in a certain town doubles every 5 years. Approximately how many years will it take for this town’s population to grow from 100 to 25000?

Detailed Solution for Test: Exponents/Powers - Question 7

To solve this problem, we need to determine the number of time intervals it takes for the population to double.

The population doubles every 5 years, so we can set up the following equation:

100 * 2n = 25000

Dividing both sides by 100, we get:

2n = 250

To solve for n, we can take the logarithm (base 2) of both sides:

n = log2(250)

Using a calculator, we find that log2(250) is approximately 7.97.

Since n represents the number of 5-year intervals, we round up to the nearest whole number to get 8.

Therefore, it will take approximately 8 * 5 = 40 years for the population to grow from 100 to 25000.

The correct answer is E, 40.

Test: Exponents/Powers - Question 8

If x = 1010 - 47, what is the sum of all the digit of x?

Detailed Solution for Test: Exponents/Powers - Question 8

To find the sum of the digits of a number, we need to add up each individual digit.

Given that x = 1010 - 47, we can simplify it as follows:

x = 1010 - 47
x = 963

To find the sum of the digits of 963, we add 9 + 6 + 3, which equals 18.

Therefore, the correct answer is E, 80.

Test: Exponents/Powers - Question 9

What is the units digit of 17381?

Test: Exponents/Powers - Question 10

What is the units digit of 720?

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