## Search found 2550 matches

#### On the first of the year, James invested x dollars at Proudstar bank in an account that yields

###### Problem Solving

##### Re: On the first of the year, James invested x dollars at Proudstar bank in an account that yields

There are several issues with the question. For one thing ,we don't know if the interest on the first account is simple interest or compound interest. If the account earns 2% simple interest every quarter year, it will earn 8% interest in a full year, and the answer is C. If instead the interest com...

- by Ian Stewart

Mon Aug 22, 2022 5:02 am- Forum: Problem Solving
- Topic: On the first of the year, James invested x dollars at Proudstar bank in an account that yields
- Replies:
**1** - Views:
**204**

#### What is the maximum possible length of an integer less than 600? (the length of an integer is the total number of prime

###### Problem Solving

##### Re: What is the maximum possible length of an integer less than 600? (the length of an integer is the total number of pr

We'll get the "longest" number by multiplying the smallest primes together -- in other words, we want to use as many 2's as we can. Since 2^9 = 512 is less than 600, but 2^10 is greater than 600, the answer is 9.

- by Ian Stewart

Tue Jun 28, 2022 11:05 am- Forum: Problem Solving
- Topic: What is the maximum possible length of an integer less than 600? (the length of an integer is the total number of prime
- Replies:
**1** - Views:
**244**

#### A number is said to be prime saturated if the product of all the different positive prime factors of n is less than the

###### Problem Solving

##### Re: A number is said to be prime saturated if the product of all the different positive prime factors of n is less than

The root of each answer choice is less than 10, so we're looking for an answer choice which has distinct prime factors that multiply to something less than 10. But the only two distinct primes that multiply to something less than 10 are 2 and 3, so we want the answer that is only divisible by 2 and ...

- by Ian Stewart

Tue Jun 28, 2022 11:02 am- Forum: Problem Solving
- Topic: A number is said to be prime saturated if the product of all the different positive prime factors of n is less than the
- Replies:
**1** - Views:
**207**

##### Re: OG 13 #229

The fraction will equal zero when the numerator (x+2)(x+3) is zero, so when x = -2 or -3, for two values of x. The numerator (x+2)(x+3) is the product of two consecutive integers. If two consecutive integers are both nonzero, they have the same sign. So the numerator will always be positive when x i...

- by Ian Stewart

Sat Mar 19, 2022 3:28 am- Forum: Problem Solving
- Topic: OG 13 #229
- Replies:
**9** - Views:
**6243**

#### A company will select 2 of the 6 candidates available to work in 2 different positions in

###### Problem Solving

##### Re: A company will select 2 of the 6 candidates available to work in 2 different positions in

The wording isn't clear enough, but if the two Tech positions are different (e.g. one is a Manager and the other an Engineer), then order matters, and we'll have 6 choices for the first position, 5 for the second, and 6*5 = 30 choices in total. If the HR positions are the same, order doesn't matter,...

- by Ian Stewart

Sat Mar 19, 2022 3:13 am- Forum: Problem Solving
- Topic: A company will select 2 of the 6 candidates available to work in 2 different positions in
- Replies:
**1** - Views:
**344**

##### Re: Rank the following quantities in order, from smallest to biggest.

2 √2 = 2 * 2^(1/2) = 2^(3/2), so we can rewrite item II: (2 √2)^35 = (2^3/2)^35 = 2^105/2 and since 105/2 < 60, this is less than 2^60. To compare 2^60 and 3^42, we can notice that 2^60 = (2^3)^20 = 8^20, and 3^42 = (3^2)^21 = 9^21, and since 8^20 has both a smaller base and smaller exponent than 9^...

- by Ian Stewart

Fri Feb 11, 2022 2:59 pm- Forum: Problem Solving
- Topic: Rank the following quantities in order, from smallest to biggest.
- Replies:
**1** - Views:
**348**

#### A shopkeeper offers two successive discounts of \(20\%\) each on a sweater and still makes a profit of \(60\%.\) By what

###### Problem Solving

##### Re: A shopkeeper offers two successive discounts of \(20\%\) each on a sweater and still makes a profit of \(60\%.\) By

When a price is reduced by 20%, it becomes 80%, or 0.8, of its original value. So if a price is discounted by 20% twice, it becomes (0.8)(0.8) = 0.64 of its original value. So if the pre-discount price of the sweater was $100, the post-discount price was $64. If the retailer still makes a 60% profit...

- by Ian Stewart

Wed Dec 01, 2021 5:07 am- Forum: Problem Solving
- Topic: A shopkeeper offers two successive discounts of \(20\%\) each on a sweater and still makes a profit of \(60\%.\) By what
- Replies:
**1** - Views:
**271**

#### During a clearance sale, a retailer discounted the original price of its TVs by \(25\%\) for the first two weeks of the

###### Problem Solving

##### Re: During a clearance sale, a retailer discounted the original price of its TVs by \(25\%\) for the first two weeks of

You can just imagine a TV costs $100. After the initial 25% discount, the TV costs $75, and if this new price is discounted by 20%, or by $15, the final price is $60, which is 60% of the original price.

- by Ian Stewart

Wed Dec 01, 2021 5:03 am- Forum: Problem Solving
- Topic: During a clearance sale, a retailer discounted the original price of its TVs by \(25\%\) for the first two weeks of the
- Replies:
**1** - Views:
**261**

##### Re: What is the value of \(5+4\cdot 5+4\cdot 5^2+4\cdot 5^3+4\cdot 5^4+4\cdot 5^5?\)

You can also replace "4" with "5 - 1", so the expression becomesGmat_mission wrote: ↑Sun Nov 28, 2021 6:51 amWhat is the value of \(5+4\cdot 5+4\cdot 5^2+4\cdot 5^3+4\cdot 5^4+4\cdot 5^5?\)

5 + (5-1)*5 + (5-1)*5^2 + (5-1)*5^3 + (5-1)*5^4 + (5-1)*5^5

= 5 - 5 + 5^2 - 5^2 + 5^3 - 5^3 + 5^4 - 5^4 + 5^5 - 5^5 + 5^6

= 5^6

- by Ian Stewart

Wed Dec 01, 2021 5:00 am- Forum: Problem Solving
- Topic: What is the value of \(5+4\cdot 5+4\cdot 5^2+4\cdot 5^3+4\cdot 5^4+4\cdot 5^5?\)
- Replies:
**2** - Views:
**373**

##### 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:
**283**

#### 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:
**261**

#### 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:
**270**

##### 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:
**281**

#### 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:
**289**

#### 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:
**270**