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Re: How many even divisors of 1600 are not multiples of 16?

We can prime factorize: 1600 = 16*100 = (2^4)(2^2)(5^2) = 2^6 * 5^2 Any divisor of 1600 thus has to look like (2^a)(5^b), where a is between 0 and 6 inclusive, and b is between 0 and 2 inclusive. If our divisor must be even, we must have at least one '2', so a must be at least 1. If our divisor is n...

by Ian Stewart

Mon Jun 14, 2021 4:09 am
Forum: Problem Solving
Topic: How many even divisors of 1600 are not multiples of 16?
Replies: 1
Views: 113

Re: A three-digit positive integer is chosen at random. What is the probability that the product of its digits is even?

The product of the digits will only be odd if all three digits are odd. If we want all of our digits to be odd, we'd have 5 choices for each digit, so there will be 5^3 = 125 such three-digit numbers. For the remaining 900 - 125 = 775 three-digit numbers, the product of the digits will be even. So i...

Re: Artificial intelligence computer HAL \(9000\) randomly picks three distinct integers from between \(1\) and \(9000\)

If you pick any three different numbers, there will be 3! = 6 different orders you can put them in. So if you pick three different numbers one at a time, the probability they will specifically be in decreasing order, one of the six possible orders, is 1/6.

Re: How many distinct prime divisors does a positive integer \(n\) have?

If n is a positive integer, 2n is clearly divisible by 2. If, as Statement 1 says, 2n has only one prime divisor, that prime divisor must be 2. But then n can be 1, 2, 2^2, 2^3 or any other power of 2. Since it is possible n = 1, it is possible n has no prime divisors, and if n is any other power of...

Re: A group of 7 students took a test. In the test, one student scored 100% and 2 students scored 0%. If the median scor

Since we have an odd number of scores, the median must be one of the test scores. So we know two of the scores are zero, one is 20, and one is 100. We don't know three of the scores. Writing all seven scores, using unknowns, in increasing order (some of the middle scores might equal each other) we h...

Re: If \(N\) is the product of all integers from \(1\) to \(100,\) both inclusive, then what is the remainder when

Between 1 and 100, there are 14 multiples of 7 (since 7*14 = 98), so 100! is divisible by at least 7^14. But two of those multiples of 7, namely 49 and 98, are multiples of 7^2, so they give us one extra 7, and 100! is actually divisible by 7^16. So 100! is a multiple of 7^16, and 100! + 100 is thus...

Re: If quadrilateral ABCD, pictured above, has perimeter of 64, what is the length of the line segment AD?

I've never needed to know, in more than ten years of answering official GMAT Quant questions, that when a rectangle's diagonals are perpendicular, that rectangle is a square. But that's what this question is testing. If you know that, Statement 2 is clearly sufficient alone, and Statement 1 is not (...

Re: A certain customer at a health food store purchases organic bananas at a price of $0.7 each, and conventional banana

If the person bought x organic and y conventional bananas, he spent 70x + 60y cents. From Statement 1, we know 70x + 60y = 560, so 7x + 6y = 56. Here 7x and 56 are both divisible by 7, so 6y must be too (if it's not clear why, we can rewrite the equation 6y = 56 - 7x, and now since the right side is...

Re: Word Problems

Ralph is giving out Valentine's Day cards to his friends. Each friend gets the same number of cards and no cards were leftover. If each friend gets at least one card, was the number of cards received by each friend more than one? 1) Ralph has 40 Valentine's Day cards to give out 2) If the number of...

by Ian Stewart

Thu Jun 10, 2021 4:56 am
Forum: Data Sufficiency
Topic: Word Problems
Replies: 1
Views: 130

Re: Word Problems

Using Statement 1 alone, we know if we double Brandon's age and double Carla's age, we get two numbers that are 4 apart. When you double two positive numbers, you double their difference, so Brandon's age and Carla's age must be 2 apart, and Statement 1 is sufficient alone. Or you could see that alg...

by Ian Stewart

Thu Jun 10, 2021 4:44 am
Forum: Data Sufficiency
Topic: Word Problems
Replies: 1
Views: 114

Re: In the figure above, \(ABCD\) is a parallelogram, and \(E\) is the midpoint of side \(AD.\) The area of triangular r

If we take the horizontal side AD to be the base b of the parallelogram, then its area is bh, where h is its corresponding height. Using the horizontal side as the base of the triangle, the triangle then has the same height h as the parallelogram. The triangle's base is b/2, because E is the midpoin...

Re: At a certain university, there are s students, \(w\) of whom are female and m of whom are male. The number of female

It's just a weighted average or mixtures situation. We know 12% of one group (women) and 25% of another group (men) are in physics. So overall, when we look at men and women together, somewhere between 12% and 25% are in physics. So the number of physics students p is somewhere between 12% of all st...

Re: Alex deposited \(x\) dollars into a new account that earned \(8\) percent annual interest, compounded annually. One

In the first year, the initial deposit of $x earns 8% interest, so after one year, the account holds 1.08x dollars. Then an additional $x is deposited in the account, so the account now contains 1.08x + x = x(1.08 + 1) dollars. This amount now earns 8% interest over the second year, so at the end of...

Re: A rectangular solid brick of iron is melted and shaped into a cube. If the areas of different sides of the brick wer

We have a rectangular block measuring L by W by H, and we know: LW = 54 LH = 36 WH = 24 Notice if we multiply all three of LW, LH and WH together, we get (LW)(LH)(WH) = (54)(36)(24) (L^2 W^2 H^2) = (6)(9)(6^2)(6)(4) (LWH)^2 = (2^2)(6^4)(3^2) LWH = 2*6^2*3 = 6^3 So the volume of the rectangular block...

Re: In the trapezoid above with height \(x,\) the sides with measures \(y\) and \(z\) are parallel. What is the area of

One way to find the area of a trapezoid is to average the lengths of the parallel sides, then multiply that average by the height. Applying that here, the area we want to find is [ (z +y)/2 ] * x and now in Statement 1, if we multiply both sides by x, and divide both sides by 2, the left side will l...