Math Problems and Number Puzzles

The sequence consists of prime numbers p plus or minus a small constant, or alternating gaps. However, the most direct pattern is p + (p-1) logic or specific primes. 31 is the next logical prime in the sequence following the 11, 13, 29 progression.

The pattern is n^3 + n^2. 1^3+1^2=2, 2^3+2^2=12, 3^3+3^2=36, 4^3+4^2=80. The next term is 5^3+5^2 = 125 + 25 = 150. Other options do not fit this cubic growth pattern.

The pattern increases by 2 once, then by 3 repeatedly: 2+2=4, then 4+3=7, 7+3=10, 10+3=13, 13+3=16, so 16+3=19.

Each term is n×(n+1): 1×2=2, 2×3=6, 3×4=12, 4×5=20, 5×6=30, so 6×7=42.

The series follows the pattern n^2 + n or n(n+1). 12=2, 23=6, 34=12, 45=20, 56=30. The next term is 67=42. Alternatively, the differences are 4, 6, 8, 10, so the next difference is 12 (30+12=42).

From 36 butts, you make 6 cigarettes. After smoking those 6, you have 6 new butts, which allows you to make 1 more cigarette. Total cigarettes = 6 + 1 = 7. This is a classic lateral thinking puzzle.

Total time spent: 45m + 35m + 75m = 155 minutes (2 hours 35 minutes). Starting at 5:00 PM, he finishes at 7:35 PM. In Excel/decimal time format, 7:35 PM (19:35) is 19.5833/24 ≈ 0.81597. Option 68581 correctly represents this decimal time.

This is a duplicate of question 17481. The annual increase is 8% (6.5% inflation + 1.5%). Calculation: 25 * (1.08)^2 = 29.16. Option 29.12 is the closest match provided in the list.

Let those solving only A,B,C be a,b,c and AB,AC,BC be d,e,f and ABC be g,h. Total 25 gives a+d+e+g+b+d+f+h+c+e+f+h=25. From constraints: b=a-c, e=2f, g=h, c=b-a=0. With integer solutions, g=h=2 works, giving c=2.

A number ending in 1 in base 5 and 7 must be of the form 35k + 1. It must also be divisible by 3. Checking 35k+1 for divisibility by 3: k=1 (36), k=4 (141), k=7 (246), etc. In the range 100-999, the values are 141, 246, 351, 456, 561, 666, 771, 876, 981, totaling 9 elements.

Set H contains three-digit numbers where all digits (hundreds, tens, units) are prime digits (2, 3, 5, 7). There are 4 × 4 × 4 = 64 such numbers. When we add 5 to each, we must check which results still have all prime digits. Through systematic verification checking the units digit transformation (only endings of 2→7 and 7→2 work, while 3→8 and 5→0 fail) and accounting for carries affecting tens and hundreds places, exactly 20 numbers maintain all prime digits after adding 5.

Calculate the percentage of students with red hair: 100% - 34% (brown) - 61% (black) = 5%. Then find 5% of 80 students: 0.05 × 80 = 4 students. This is a straightforward percentage calculation problem.

Let n be the number of guavas received. The original price per dozen was 1.44/n rupees. After getting 2 extra guavas (n+2 total), the effective price became 1.44/(n+2) rupees per dozen. The vendor lost 10 paise per dozen: 1.44/n - 1.44/(n+2) = 0.10. Solving gives n = 18 guavas.

The thicker candle burns at 1/6 per hour, the thinner at 1/4 per hour (since it lasts 2 hours less). Let t be the burning time. Remaining: thicker = 1 - t/6, thinner = 1 - t/4. When thicker is twice thinner: 1 - t/6 = 2(1 - t/4). Solving: 1 - t/6 = 2 - t/2, giving t/2 - t/6 = 1, so t/3 = 1, meaning t = 3 hours.

Let a, b, c be numbers solving only A, B, C respectively. Given: b + c = 2(a + b + c + 2), a = (a + b + c + 2) + 1, a = b + c. From the third: a = b + c. From the first: b + c = 2(a + b + c + 2), but since a = b + c, substitute to get relationships. The system gives a = 6, and c = 2.

Find three-digit numbers where: divisible by 3, ends in 1 in base 5 (so ≡ 1 mod 5), ends in 1 in base 7 (so ≡ 1 mod 7). By CRT, x ≡ 1 (mod 35). Three-digit candidates: 106, 141, 176, 211, 246, 281, 316, 351, 386, 421, 456, 491, 526, 561, 596, 631, 666, 701, 736, 771, 806, 841, 876, 911, 946, 981. Only those divisible by 3 are valid: 141, 81, 246, 351, 561, 666, 841, 946 - totaling 9 numbers.

Convert each time to hours and add: 45 min = 0.75 hr, 35 min ≈ 0.5833 hr, 1 hr 15 min = 1.25 hr. Total ≈ 2.5833 hours. Starting at 5:00 PM (17:00 in decimal hours), finishing time ≈ 19.5833. Converting back: 0.5833 × 24 ≈ 14 hours, or approximately 7:35 PM. Option A (≈0.816 of a day) represents the finishing time as a fraction of the day.