Find the least positive integer m such that 2^2.3^5.7.11.m is a perfect square.

Fatimamalik

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From this 2010 thread:

Suppose that in standard factored form a = p1^e1 * p2^e2 ...* pk^ek, where k is a positive integer; p1, p2,...,pk are prime numbers; and e1, e2,...,ek, are positive integers.

a) What is the standerd factored form for a^2?

I know the answer to this is p1^2*e1 * p2^2*e2... * pk^2*ek.

b) Find the least posivtive integer n such that 2^5 * 3 * 5^2 * 7^3 * n is a perfect square. Write the resulting product as a perfect square.

n = 42, 2^5 * 3 * 5^2 * 7^3 * n = 5880^2

The answer to this is in the back of the book....
For the same question there is a part c for which im confused

C. Find the least positive integer m such that 22.35.7.11.m is a perfect square. Write the resulting product as a perfect square
 
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First off, I'm assuming the decimal points are intended to be multiplication symbols, and thus it's actually: 23 * 35 * 7 * 11 * m. If that assumption is correct, then think about what the problem is asking. You have several constants and a variable. What if you multiplied all of the constants together? And then you know you want to find an m such that the expression equals a perfect square. Well, a perfect square can be written as some integer n squared, right? So, what if you take the square root of both sides?

149688m=n2\displaystyle 149688m=n^2

149688m=n\displaystyle \sqrt{149688m}=n

You know that both m and n have to be integers, so what is the smallest m that fulfills those conditions?
 
From this 2010 thread:


For the same question there is a part c for which im confused

C. Find the least positive integer m such that 22.35.7.11.m is a perfect square. Write the resulting product as a perfect square

m=3711\displaystyle \large m=3\cdot 7\cdot 11 WHY?
 
From this 2010 thread:


For the same question there is a part c for which im confused

C. Find the least positive integer m such that 22.35.7.11.m is a perfect square. Write the resulting product as a perfect square
One way to look at questions such as this [perfect square, perfect cube, etc.] is to consider just what it means and the prime factorization of the number. For example, for squares, a is a perfect square if a1/2 is an integer (implying also that a is an integer). That means there is some integer b such that
b2 = a
Now if b is factored into prime factors p1, p2, p3, ... then
b = p1e1p2e2p3e3...,pnen\displaystyle p_1^{e_1}\, *\, p_2^{e_2}\, *\, p_3^{e_3}\, ...,\, *\, p_n^{e_n}
where the ej are integers, or squaring b,
b2 = p12e1p22e2p32e3...,pn2en\displaystyle p_1^{2\, e_1}\, *\, p_2^{2\, e_2}\, *\, p_3^{2\, e_3}\, ...,\, *\, p_n^{2\, e_n}.
Thus all the exponents of the prime decomposition of a must be even.

Also note that integer in this context means a positive integer.
 
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