What is the units digit of 17^{27}?
What is the remainder when 2^{243} is divided by 10?
How many terminating zeroes does 200! have?
If x is a positive integer, what is the units digit of x^{2}?
1) The units digit of x^{4} is 1
2) The units digit of x is 3
If p is a positive integer, and x = m^{1/3}, y = n^{1/2}, and z = 2p, then which one of x, y, and z is the greatest?
1) 4p^{3} = 5m
2) 5n = 3p^{2 }
Is a^{x} equal to 4?
1) a^{x + 1} = 4
2) (a + 1)^{x} = 4
How many different positive integers exist between 10^{6} and 10^{7}, the sum of whose digits is equal to 2?
Method 1 to solve this GMAT Number Properties Question: Find the number of such integers existing for a lower power of 10 and extrapolate the results.
Between 10 and 100, that is 10^{1} and 10^{2}, we have 2 numbers, 11 and 20.
Between 100 and 1000, that is 10^{2} and 10^{3}, we have 3 numbers, 101, 110 and 200.
Therefore, between 10^{6} and 10^{7}, one will have 7 integers whose sum will be equal to 2.
Alternative approach
All numbers between 10^{6 }and 10^{7} will be 7 digit numbers. There are two possibilities if the sum of the digits has to be '2'.
Possibility 1: Two of the 7 digits are 1s and the remaining 5 are 0s. The left most digit has to be one of the 1s. That leaves us with 6 places where the second 1 can appear.
So, a total of six 7digit numbers comprising two 1s exist, sum of whose digits is '2'.
Possibility 2: One digit is 2 and the remaining are 0s. The only possibility is 2000000. Total count is the sum of the counts from these two possibilities = 6 + 1 = 7
If = a, what is the units digits of ?
Take common (13!^8) from numerator
take common (13!^4) from denominator
now 13! has last digit 0
so 01 =1=10+(1)=9
Hence Option D is correct
Is x > 10^{10}?
1) x > 2^{34}
2) x = 2^{35}
If x = 23^{2} * 25^{4} * 27^{6} * 29^{8} and x is a multiple of 26^{n} where n is a nonnegative integer, then what is the value of n^{26} – 26^{n}?
If x is a positive integer, what is the remainder when 7^{12x + 3} + 3 is divided by 5?
In which of the following choices must p be greater than q?
What is the greatest prime factor of 4^{17} – 2^{28}?
How many keystrokes are needed to type numbers from 1 to 1000?
While typing numbers from 1 to 1000, there are 9 single digit numbers: from 1 to 9.
Each of these numbers requires one keystroke.
That is 9 key strokes.
There are 90 twodigit numbers: from 10 to 99.
Each of these numbers requires 2 keystrokes.
Therefore, 180 keystrokes to type the 2digit numbers.
There are 900 threedigit numbers: from 100 to 999.
Each of these numbers requires 3 keystrokes.
Therefore, 2700 keystrokes to type the 3digit numbers.
1000 is a fourdigit number which requires 4 keystrokes.
Totally, therefore, one requires 9 + 180 + 2700 + 4 = 2893 keystrokes.
Positive integers a, b, c, m, n, and p are defined as follows: m = 2^{a}3^{b}, n = 2^{c}, and p = 2m/n, is p odd?
1) a < b
2) a < c
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