## Search found 2541 matches

##### 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**

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

###### Problem Solving

##### 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...

- by Ian Stewart

Sun Jun 13, 2021 2:03 pm- Forum: Problem Solving
- Topic: A three-digit positive integer is chosen at random. What is the probability that the product of its digits is even?
- Replies:
**1** - Views:
**107**

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

###### Problem Solving

##### 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.

- by Ian Stewart

Sat Jun 12, 2021 4:26 pm- Forum: Problem Solving
- Topic: Artificial intelligence computer HAL \(9000\) randomly picks three distinct integers from between \(1\) and \(9000\)
- Replies:
**1** - Views:
**106**

##### 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...

- by Ian Stewart

Sat Jun 12, 2021 3:59 pm- Forum: Data Sufficiency
- Topic: How many distinct prime divisors does a positive integer \(n\) have?
- Replies:
**1** - Views:
**110**

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

###### Data Sufficiency

##### 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...

- by Ian Stewart

Sat Jun 12, 2021 3:55 pm- Forum: Data Sufficiency
- Topic: A group of 7 students took a test. In the test, one student scored 100% and 2 students scored 0%. If the median score of
- Replies:
**1** - Views:
**114**

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

###### Problem Solving

##### 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...

- by Ian Stewart

Sat Jun 12, 2021 3:43 pm- Forum: Problem Solving
- Topic: If \(N\) is the product of all integers from \(1\) to \(100,\) both inclusive, then what is the remainder when
- Replies:
**1** - Views:
**96**

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

###### Data Sufficiency

##### 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 (...

- by Ian Stewart

Thu Jun 10, 2021 5:04 am- Forum: Data Sufficiency
- Topic: If quadrilateral ABCD, pictured above, has perimeter of 64, what is the length of the line segment AD?
- Replies:
**1** - Views:
**112**

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

###### Data Sufficiency

##### 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...

- by Ian Stewart

Thu Jun 10, 2021 5:00 am- Forum: Data Sufficiency
- Topic: A certain customer at a health food store purchases organic bananas at a price of $0.7 each, and conventional bananas at
- Replies:
**1** - Views:
**113**

##### 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**

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

###### Problem Solving

##### 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...

- by Ian Stewart

Tue Jun 08, 2021 5:20 am- Forum: Problem Solving
- Topic: In the figure above, \(ABCD\) is a parallelogram, and \(E\) is the midpoint of side \(AD.\) The area of triangular regio
- Replies:
**1** - Views:
**133**

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

###### Problem Solving

##### 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...

- by Ian Stewart

Tue Jun 08, 2021 5:13 am- Forum: Problem Solving
- Topic: At a certain university, there are s students, \(w\) of whom are female and m of whom are male. The number of female phy
- Replies:
**1** - Views:
**127**

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

###### Problem Solving

##### 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...

- by Ian Stewart

Tue Jun 08, 2021 5:03 am- Forum: Problem Solving
- Topic: Alex deposited \(x\) dollars into a new account that earned \(8\) percent annual interest, compounded annually. One year
- Replies:
**1** - Views:
**122**

#### A rectangular solid brick of iron is melted and shaped into a cube. If the areas of different sides of the brick were 24

###### Problem Solving

##### 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...

- by Ian Stewart

Tue Jun 08, 2021 4:57 am- Forum: Problem Solving
- Topic: A rectangular solid brick of iron is melted and shaped into a cube. If the areas of different sides of the brick were 24
- Replies:
**1** - Views:
**122**

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

###### Data Sufficiency

##### 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...

- by Ian Stewart

Tue Jun 08, 2021 4:27 am- Forum: Data Sufficiency
- Topic: In the trapezoid above with height \(x,\) the sides with measures \(y\) and \(z\) are parallel. What is the area of the
- Replies:
**1** - Views:
**107**