Is the last digit of integer x^{2} – y^{2} a zero?
1) x – y is an integer divisible by 30
2) x + y is an integer divisible by 70
How many different prime numbers are factors of the positive integer n?
1) Four different prime numbers are factors of 2n
2) Four different prime numbers are factors of n^{2}
If k is an integer greater than 1, is k equal to 2^{r} for some positive integer r?
1) k is divisible by 2^{6}
2) k is not divisible by any odd integer greater than 1
If 500 is the multiple of 100 that is closest to X and 400 is the multiple of 100 closest to Y, then which multiple of 100 is closest to X + Y?
1) X < 500
2) Y < 400
If the number x3458623y is divisible by 88, what is the value of x?
Is m divisible by n?
1) m + n is divisible by m – n
2) m + n is divisible by n
Is r a multiple of s?
1) r + 2s is a multiple of s
2) 2r + s is a multiple of s
What is the greatest prime factor of 2^{100} – 2^{96}?
If n and k are positive integers, is n divisible by 6?
(1) n = k(k+1)(k1)?
(2) k – 1 is a multiple of 3.
A positive integer n is said to be “primesaturated” if the product of all the different positive prime factors of n is less than the square root of n. What is the greatest twodigit primesaturated integer?
How many different factors does the integer n have?
1) n = a^{4}b^{3}, where a and b are different positive prime numbers. ?
2) The only positive prime numbers that are factors of n are 5 and 7. ?
If the integer k is a multiple of 3, which of the following is also a multiple of 3?
If x and y are nonzero integers, is 18 a factor of xy^{2}?
1) x is a multiple of 54. ?
2) y is a multiple of 6. ?
If both 11^{2} and 3^{3} are factors of the number a * 4^{3} * 6^{2} * 13^{11}, then what is the smallest possible value of 'a'?
Step 1 of solving this GMAT Number Properties Question: Prime factorize the given expression
a * 4^{3} * 6^{2} * 13^{11} can be expressed in terms of its prime factors as a * 2^{8} * 3^{2} * 13^{11}
Step 2 of solving this GMAT Number Properties Question: Find factors missing after excluding 'a' to make the number divisible by both 11^{2} and 3^{3}
11^{2} is a factor of the given number.
If we do not include 'a', 11 is not a prime factor of the given number.
If 11^{2} is a factor of the number, 11^{2} should be a part of 'a'
3^{3} is a factor of the given number.
If we do not include 'a', the number has only 3^{2} in it.
Therefore, if 3^{3} has to be a factor of the given number 'a' has to contain 3^{1} in it.
Therefore, 'a' should be at least 11^{2} * 3 = 363 if the given number has 11^{2} and 3^{3} as its factors.
The question is "what is the smallest possible value of 'a'?"
The smallest value that 'a' can take is 363
Choice C is the correct answer.
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
Choice B is the correct answer.
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