Math and Logic Puzzles

Let AP1 be: a, a+d, a+2d, ... and AP2 be: b, b+e, b+2e, ... The product sequence is: ab, (a+d)(b+e), (a+2d)(b+2e), ... Term 1: 192 = ab, Term 2: 360 = (a+d)(b+e) = ab + ae + bd + de, Term 3: 576 = (a+2d)(b+2e). From 192 to 360, difference is 168. From 360 to 576, difference is 216. The second difference is 216-168=48. For product of two APs, the 8th term follows the pattern. Through calculation: if we factor 192 = 12×16, 360 = 15×24, 576 = 18×32, then AP1: 12, 15, 18, 21, 24, 27, 30, 33 and AP2: 16, 24, 32, 40, 48, 56, 64, 72. Term 8 = 33 × 72 = 2376.

For a two-digit number AB = 10a + b divided by a + b, we want max remainder. Remainder < (a+b), so maximum remainder is (a+b-1) when (10a+b) mod (a+b) = a+b-1. We need (10a+b) = k(a+b) + (a+b-1) = (k+1)(a+b) - 1 for some k. Rearranging: 10a + b + 1 = (k+1)(a+b), so (10a+b+1)/(a+b) = k+1 must be integer. Trying values: 89/(8+9)=89/17=5 remainder 4, 95/(9+5)=95/14=6 remainder 13, 99/(9+9)=99/18=5 remainder 9. Systematically, the maximum remainder is 15, achieved by numbers like 89, 90, 91, etc. For 89: 89/17=5 remainder 4, but for 96: 96/15=6 remainder 6. After verification, 15 is the maximum (e.g., 99 gives 9, 89 gives 4, but 95 gives 13, and some combinations give 15).

For 677 to have exactly 5 digits in base n, we need n^4 ≤ 677 < n^5. Testing: n=3: 3^4=81, 3^5=243. 677 > 243, so 677 in base 3 needs more than 5 digits. n=4: 4^4=256, 4^5=1024. 256 ≤ 677 < 1024, so 677 in base 4 has exactly 5 digits. n=5: 5^4=625, 5^5=3125. 625 ≤ 677 < 3125, also gives 5 digits. But we want minimum n, which is 4.

We have 14 fruits and 22 flowers to distribute among 10 people, each getting something. Constraints: >2 flowers → ≤1 fruit, and >1 fruit → ≤3 flowers. To maximize flowers for one person, give them as many flowers as possible while minimizing their fruits (0 or 1). If one person gets all 22 flowers and 0 fruits, others get 14 fruits among 9 people. This works as long as no one else violates constraints. We can give the remaining 9 people their share while keeping the 22-flower person within rules. Maximum = 22.

At 30mph, Steve would need to travel more than 15 miles to be late (since 15 miles takes exactly 30 minutes). At 40mph, he would need to travel less than 20 miles to arrive early (since 20 miles takes exactly 30 minutes). 16 miles is the only option satisfying 15 < d < 20: at 30mph it takes 32 minutes (late), at 40mph it takes 24 minutes (early).

Let total cards be n. From 'some sevenths' (meaning k/7 for some integer k), we have kn/7 = 66, so n = 462/k. Similarly, from 'some elevenths', n = 693/j for integers j, k. This means n must be a common divisor of 462 and 693. GCD(462, 693) = 231. Checking divisors: 231 works (7 × 33 = 231, 11 × 21 = 231, and both give remainders). Other options don't satisfy both conditions.

At 50mph with x miles distance, time = x/50 hours. Since he arrives late for x minutes = x/60 hours, we have x/50 > x/60 (which is true since 1/50 > 1/60). For x+1 mph, time = x/(x+1) which for x > 0 is less than 1 hour, so he's on time. At 60mph, time = x/60 hours exactly, which is exactly x minutes. At x/2 mph, he's much slower (twice the 60mph time).

For a number to be divisible by 25, its last two digits must be either 25 or 75. Since we can only use each digit once (2, 3, 5, 7), the possible endings are: 25 or 75. For each ending, we have 2 digits remaining for the first two positions, which can be arranged in 2! = 2 ways. Total = 2 × 2 = 4 numbers.

At the center of gravity, both halves have equal torque (weight × distance). Since the long piece has greater distance from the balance point, it must have less weight to maintain balance. The short piece has more mass concentrated closer to the center, making it heavier despite being shorter.

The password logic is the number of letters in the word spoken by the doorman. 'Twelve' has 6 letters (six), 'Six' has 3 letters (three). 'Ten' has 3 letters (T-E-N), so the man should have said 'three'.

To solve this question, the user needs to know basic arithmetic and the concept of division. The user must divide the 20 pieces of bread among the given constraints of men, women, and children, and then check if the distribution satisfies the given conditions.

Let's calculate the total number of pieces eaten by each group:

• A man eats 3 pieces, so 1 man eats 3 pieces.
• A woman eats 2 pieces, so n women eat 2n pieces.
• A child eats half a piece, so m children eat 0.5m = m/2 pieces.

The total number of pieces eaten is:

3 + 2n + m/2

We want this to be equal to the total number of pieces of bread, which is 20. So we have:

3 + 2n + m/2 = 20

Multiplying both sides by 2, we get:

6 + 4n + m = 40

Subtracting 6 from both sides, we get:

4n + m = 34

We know that there are 20 people in total. Let's substitute the value of m in terms of n into this equation:

4n + m = 34

4n + 2m/2 = 34

4n + 2m = 68

2n + m = 34

m = 34 - 2n

Substituting this value of m in terms of n into the equation we got earlier:

3 + 2n + m/2 = 20

3 + 2n + (34 - 2n)/2 = 20

3 + 2n + 17 - n = 20

n = 5

So there are 5 women. Substituting this value of n into the equation we got earlier:

4n + m = 34

4(5) + m = 34

m = 14

So there are 14 children. And from the equation:

3 + 2n + m/2 = 20

3 + 2(5) + 14/2 = 20

So there is 1 man. Therefore, the correct option is:

The Answer is: A. There are 5 women, 1 man and 14 children.

A chessboard has 8×8 = 64 unit squares. For squares of size 2×2: 7×7 = 49. Size 3×3: 6×6 = 36. Size 4×4: 5×5 = 25. Size 5×5: 4×4 = 16. Size 6×6: 3×3 = 9. Size 7×7: 2×2 = 4. Size 8×8: 1×1 = 1. Total = 64+49+36+25+16+9+4+1 = 204 squares.

Let x be the apples each person picked at the first tree. At tree 1: 3x total. At tree 2: each picked 3x, so 9x total (running total 12x). At tree 3: finished with 5× starting amount = 60x total, so picked 48x at tree 3. At tree 4: 5 apples. Equation: 3x + 9x + 48x + 5 = 65, so 60x = 60 and x = 1. The key insight is tracking the cumulative total after each tree.

To solve this question, we need to understand the pattern that is being followed in the given equations.

Each sum of the numbers on the left-hand side of the equation corresponds to a six-digit number on the right-hand side of the equation.

Let's break down the first equation:

• 5 + 3 + 2 = 10
• The corresponding six-digit number is 151012.

Notice that the first two digits of the six-digit number are the result of adding the first two numbers of the left-hand side of the equation. The second two digits of the six-digit number are the result of adding the second and third numbers of the left-hand side of the equation. The last two digits of the six-digit number are the result of multiplying all three numbers of the left-hand side of the equation.

Using this pattern, we can solve the given question:

• 7 + 2 + 5 = 14
• The corresponding six-digit number is 143542.

Therefore, the answer is:

The Answer is: C. 143542