/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 5 A perfect square is a positive i... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 5

A perfect square is a positive integer which when square rooted results in an integer. If \(N=3^4 \cdot 5^3 \cdot 7\), then what is the biggest perfect square that is a factor of \(N\) ? (A) \(3^2\) (B) \(5^2\) (C) \(9^2\) (D) \((9 \cdot 5)^2\) (E) \((3 \cdot 5 \cdot 7)^2\)

Short Answer

Expert verified
The biggest perfect square factor of \( N \) is (9路5)虏 which is 2025.

Step by step solution

01

Understand the Problem

We are given a number \( N = 3^4 \cdot 5^3 \cdot 7 \) and need to find the largest perfect square that divides \( N \). A perfect square has all even powers in its prime factorization.
02

Factorize N into Primes

The prime factorization of \( N \) is already given as \( 3^4 \cdot 5^3 \cdot 7 \). Our task is to construct a factor of \( N \) that is a perfect square, i.e., each prime's exponent must be even.
03

Adjust Exponents to be Even

For the number to be a perfect square, adjust the exponents: - \( 3^4 \) stays as is since 4 is even.- \( 5^3 \) becomes \( 5^2 \) because 2 is the largest even number less than 3.- \( 7^1 \) becomes \( 7^0 \) (which means it isn't included) because 0 is even and less than 1.
04

Construct the Largest Perfect Square Factor

The largest perfect square factor of \( N \) is \( 3^4 \cdot 5^2 \cdot 7^0 = 3^4 \cdot 5^2 \). Compute this as follows: - \( 3^4 = 81 \)- \( 5^2 = 25 \)So the factor becomes: \( 81 \cdot 25 = 2025 \) which is equivalent to \((9 \cdot 5)^2\) as \((9 \cdot 5)^2 = 2025\).
05

Verify Against Options

From the given options, we see that \((9 \cdot 5)^2\) matches the calculated factor of 2025. Thus this is the correct option.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Factorization
A perfect square is a number that results from squaring an integer. That means it can be written as another integer multiplied by itself. For a number to qualify as a perfect square, all exponents in its prime factorization need to be even. The process of identifying or creating a perfect square involves ensuring that each prime number within the number has an even exponent.
  • A perfect square like 36 has a prime factorization of \(2^2 \times 3^2\).
  • Here, both exponents are even, confirming it's a perfect square.
Understanding how to factorize a number into its perfect square components is an essential skill for mastering GMAT Math.
Exponent Adjustment
Adjusting exponents is essential when transforming a number into a perfect square. When working with a number like \(N = 3^4 \cdot 5^3 \cdot 7\), you want to create a new number with all even exponents.
  • "Even" means divisible by 2, like 2, 4, 6, and so on.
  • Retain the highest even exponent close to the original exponent.
  • For example, 5 has an exponent of 3, so you adjust it down to 2 because 2 is even and the largest even number less than 3.
This adjustment ensures that the resulting factor remains a perfect square because all prime factors will have even exponents.
Prime Factorization
Prime factorization breaks down a number into its smallest prime number multipliers. In the context of perfect squares, the factorization can tell us which parts of a number need adjustment.
  • The number \(N = 3^4 \cdot 5^3 \cdot 7\) is already in its prime factorized form.
  • This form helps visualize how each constituent prime number contributes to \(N\).
  • It allows us to see which exponents need to be modified in order to achieve an even distribution.
Prime factorization is a critical step before adjusting exponents. Without it, you can't effectively analyze or modify a number to become a perfect square.
Largest Perfect Square Factor
Once you've adjusted the exponents in a number so that all are even, you have constructed the largest possible perfect square factor of that number. The formula becomes simple arithmetic after exponent adjustments.
  • With the altered exponents, our number \(N\) transformed to \(3^4 \cdot 5^2 \cdot 7^0\).
  • Calculate \(3^4 = 81\) and \(5^2 = 25\).
  • Multiply these results to obtain \(81 \times 25 = 2025\).
  • This number \(2025\) corresponds to \((9 \cdot 5)^2\).
This reflects the methodical way you combine factorization, exponent adjustment, and arithmetic to solve for the largest perfect square factor. It鈥檚 a critical calculation that鈥檚 not only helpful for GMAT Math but also valuable for broader mathematical problem-solving.

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Most popular questions from this chapter

If \(p=\frac{\sqrt{3}-2}{\sqrt{2}+1}\), then which one of the following equals \(p-4\) ? (A) \(\sqrt{3}-2\) (B) \(\sqrt{3}+2\) (C) 2 (D) \(-2 \sqrt{2}+\sqrt{6}-\sqrt{3}-2\) (E) \(\quad-2 \sqrt{2}+\sqrt{6}-\sqrt{3}+2\)

If \(x=10^{1.4}, y=10^{07}\), and \(x^z=y^3\), then what is the value of \(z\) ? (A) 0.5 (B) 0.66 (C) 1.5 (D) 2 (E) 3

$$ \frac{4(\sqrt{6}+\sqrt{2})}{\sqrt{6}-\sqrt{2}}-\frac{2+\sqrt{3}}{2-\sqrt{3}}= $$ (A) 1 (B) \(\sqrt{6}-\sqrt{2}\) (C) \(\sqrt{6}+\sqrt{2}\) (D) 8 (E) 12

In which one of the following choices must \(p\) be greater than \(q\) ? (A) \(0.9^p=0.9^a\) (B) \(0.9^p=0.9^{2 q}\) (C) \(0.9^p>0.9^q\) (D) \(9^p<9^9\) (E) \(9^p>9^q\)

$$ \frac{\frac{\left((\sqrt{7})^x\right)^2}{(\sqrt{7})^{11}}}{\frac{7^x}{7^{11}}}= $$ (A) \(\sqrt{7}\) (B) 7 (C) \(7^2\) (D) \(7^{1 / 2}\) (E) \(7^{11}\)
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