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What is the smallest positive integer n such that 2n is a perfect square and 3n is a perfect cube?
n=72 satisfies the relation. 2*72 = 12**2, and 3*72 = 6**3. The only question is whether or not that is the smallest integer that will do so.
All perfect squares of which 2n is a factor are of form (2x)**2 = 4x**2, with n= 2x**2. Similarly all perfect cubes of which 3n is a factor are of form (3y)**3 = 27y**3 with n = 9y**3.
So, we need integers x and y such that 2x**2 = 9y**3. If the integers x and y are the smallest that satisfy the equation, then we have the smallest n. It doesn't work for y = 1, or any other odd number. If y = 2, n = 72. There is a number x =6 which satisfies the relation. Since y is smallest possible, then n=72 is the smallest positive integer that satisfies the relation.
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What is the smallest positive integer that 63 can be multiplied by in order for the product to be a perfect square?
63= 9* 7. 9 is already a perfect square, so mulitiply by 7. 7 is your answer.
What is the smallest integer greater than 1 that is both a perfect square and perfect cube?
its 81
What is the smallest positive integer that 248832 can be multiplied by in order for the product to be a perfect square?
The prime factorisation of 248832 is 2¹⁰ × 3⁵ Every perfect square number has a prime factorisation where each prime is to an even power. 2 has an even power 3 has an odd power, so need an extra power → multiple 248832 by 3 which gets (2⁵ × 3³)²
What is a perfect square?
A perfect square is a square of an integer (a whole number).
What is the smallest number that divides 275 to give a perfect square?
The smallest integer is 11 but there is no smallest number! 0.11 is a smaller number and will give a perfect square. 0.0011 is smaller still, and 0.000011 even smaller. That sequence is endless!
Find the smallest positive integer k such that 1176k is a perfect square?
6.
What is the smallest positive integer value of x for which 54x is a perfect square?
324
What is the smallest integer where the integer before it is a perfect square and the integer after it is a perfect cube?
The integer is 26
What is the smallest positive integer that 63 can be multiplied by in order for the product to be a perfect square?
63= 9* 7. 9 is already a perfect square, so mulitiply by 7. 7 is your answer.
What is the smallest positive integer that has exactly 6 perfect square divisors?
the answer is 144, it is divisible by 1, 4, 9, 16, 36, and 144.
What is the smallest integer greater than 1 that is both a perfect square and perfect cube?
its 81
Are square roots of positive integers rational?
Only if the integer is a perfect square.
Is the square root of eight rational or irrational?
Irrational. The square root of a positive integer is either an integer (that is, if the integer is a perfect square), or an irrational number.
Is the square root of 39 irrational?
Yes. The square root of a positive integer can only be an integer (if your integer is a perfect square), or an irrational number (if it isn't).
What is the smallest positive integer q for which 1008q is a perfect square?
In terms of prime factors, 1008 = 24*32*7 Then since 24 and 32 are perfect squares, all that is required is to make 7q a perfect square and so q = 7.
What is the smallest positive integer that 248832 can be multiplied by in order for the product to be a perfect square?
The prime factorisation of 248832 is 2¹⁰ × 3⁵ Every perfect square number has a prime factorisation where each prime is to an even power. 2 has an even power 3 has an odd power, so need an extra power → multiple 248832 by 3 which gets (2⁵ × 3³)²
What is least positive integer by which 68 should be multiplied so that resultant integer is a perfect square?
17
