Math Logic & Pattern Puzzles

This sequence has two interleaved patterns. Odd positions (1st, 3rd, 5th, 7th, 9th): 4, 6, 8, 10, 12 - adding 2 each time. Even positions (2nd, 4th, 6th, 8th): 2, 4, 8, __ - doubling each time. The missing even position value is 8×2=16.

The rule for CHEERS→EHCSRE is: swap the first two letters (CH→HC), then reverse the remaining four letters (EERS→SRE). Applying to BASKET: swap BA→AB, reverse KET→TEK, giving ABTEK+SABTEK. Actually, re-examining: CHEERS (6 letters) becomes EHCSRE. Position 1 and 2 swap, positions 3-6 reverse. For BASKET: BA→SA (swap), KET→TEK (reverse), giving SABTEK.

This is a similar triangles problem. The pole and its shadow form a right triangle with ratio 20:5 = 4:1. The building and its shadow form a similar triangle, so the building's height is 4 times its shadow length. 4 × 20 = 80 feet. Option A (400) incorrectly multiplies 20 × 20, option B (4) uses the inverse ratio, and option D (100) has no clear mathematical basis.

The lights change simultaneously at intervals equal to the LCM of 36, 42, and 72 seconds. LCM(36,42,72) = 504 seconds = 8 minutes 24 seconds. Starting at 9:00 AM, they will change together again at 9:08:24 AM. To find LCM: 36=2^2×3^2, 42=2×3×7, 72=2^3×3^2, so LCM=2^3×3^2×7=504.

96 = 2^5 × 3, so even divisors are 2, 4, 6, 8, 12, 16, 24, 32, 48, 96 (sum = 248). 3600 = 2^4 × 3^2 × 5^2, so odd factors are powers of 3 × 5: (1+3+9) × (1+5+25) = 13 × 31 = 403. Total = 248 + 403 = 651.

Trailing zeros in n! equal the exponent of 5 in n!. For 127!: floor(127/5) + floor(127/25) + floor(127/125) = 25 + 5 + 1 = 31. This is a standard application of Legendre's formula.

For divisibility by 36 = 4 × 9: 34x5y must be divisible by 4 (so 5y divisible by 4, y ∈ {2,6}) AND divisible by 9 (sum 3+4+x+5+y = 12+x+y divisible by 9). Testing options: Only 3462 works - 62 divisible by 4, sum 3+4+6+2=15 (not divisible by 9). Wait, that fails. Let me recalculate: For 34x5y divisible by 36, need divisible by 4 and 9. y must be 2 or 6 for divisibility by 4. Sum 12+x+y must be divisible by 9. Option D (4,2): 34,4252? That doesn't make sense. The number is 34x5y, so x=4, y=2 gives 34452. Sum = 3+4+4+5+2=18 (divisible by 9). 52 is divisible by 4. Correct.

720 = 2^4 × 3^2. For 720^n to divide 77!, we need at least 4n factors of 2 and 2n factors of 3 and n factors of 5. Count of 5s in 77! is floor(77/5) + floor(77/25) = 15 + 3 = 18. Count of 3s is floor(77/3) + floor(77/9) + floor(77/27) + floor(77/81) = 25 + 8 + 2 + 0 = 35. We need 2n ≤ 35 and n ≤ 18, giving n ≤ 17 (since 2×18 = 36 > 35).

To find the maximum n such that 77! is divisible by 720^n, we factor 720 = 2^4 × 3^2 × 5. In 77!, the exponent of 2 is 73, of 3 is 35, and of 5 is 18. For 720^n = 2^(4n) × 3^(2n) × 5^n, we need 4n ≤ 73, 2n ≤ 35, and n ≤ 18. The limiting factor is 3, giving n ≤ 17 (since 35/2 = 17.5). Thus, maximum n = 17.

This sequence follows two interleaved patterns. The odd-positioned terms (1st, 3rd, 5th, 7th) are 7, 9, 11, 13 - consecutive odd numbers increasing by 2. The even-positioned terms (2nd, 4th, 6th) are 5, 3, 1 - decreasing by 2. After 1, the next term (8th position) should continue the odd number pattern at 13, making the next odd-positioned term (9th) equal to 15. Alternatively, notice the pattern alternates between -2, +4, -6, +8, -10, +12 - making the next term 15.

This is an interleaved pattern with two separate sequences. Even-positioned terms (2nd, 4th, 6th, ...): -3, -9, -15, ? follow the pattern subtracting 6 each time (-9 = -3 - 6, -15 = -9 - 6). Continuing: -15 - 6 = -21, then -21 - 6 = -27, then -27 - 6 = -33. Odd-positioned terms (1st, 3rd, 5th, ...): 4, 24, 124, 624, 3124 follow a pattern where each term is 5 times the previous odd term plus 4. The negative number after 3124 is -33.

The series follows a multiplication pattern with increments of 0.5: $4 imes 0.5 = 2$; $2 imes 1.5 = 3$; $3 imes 2.5 = 7.5$; $7.5 imes 3.5 = 26.25$. The next step is $26.25 imes 4.5 = 118.125$.

Find the pattern in differences between consecutive terms. 2 to 3: +1, 3 to 4: +1, 4 to 6: +2, 6 to 8: +2, 8 to 10: +2, 10 to 13: +3, 13 to 16: +3, 16 to 20: +4, 20 to 24: +4. The differences are 1,1,2,2,2,3,3,4,4... The next differences should be 4,5,5 giving: 24+4=28, 28+5=33, 33+5=38. This matches option D exactly.

The pattern is adding the sum of the digits of the current number to itself: $17+(1+7)=25$; $25+(2+5)=32$; $32+(3+2)=37$; $37+(3+7)=47$; $47+(4+7)=58$; $58+(5+8)=71$; $71+(7+1)=79$. Thus, the next number is 79.

This is an alternating pattern with two interleaved sequences. Odd-positioned terms (1st, 3rd, 5th, ...): 7, 9, 11, ? increase by 2 each time. Even-positioned terms (2nd, 4th, 6th): 5, 3, 1 decrease by 2 each time. Following the odd-positioned pattern: 7+2=9, 9+2=11, so 11+2=13. The answer is 13.

Profit = 10% of 50 crore = 5 crore. Fee = 10% of profit = 50 lakhs. Signing amount paid = 20% of fee = 10 lakhs. Remaining payment = 50 lakhs - 10 lakhs = 40 lakhs. Therefore, the producer owes Mallika 40 lakhs.

Displacement is defined as the straight-line distance between the starting point and the final position. Since the helicopter returned to point A and parked at the exact same spot where it took off, the total displacement is 0 km, regardless of the distance traveled.

Let x be the amount given to the beggar. Since the beggar originally had the same amount as what was donated, after receiving x, he has 2x. Your remaining money is 100-x. Setting up the ratio 2x/100-x = 2/9 gives 18x = 200-2x, so 20x = 200 and x = 10. Therefore, 90 rupees remain with you.

Work backward from 0. Person 4: $(0 + 0.5) imes 2 = 1$. Person 3: $(1 + 0.5) imes 2 = 3$. Person 2: $(3 + 0.5) imes 2 = 7$. Person 1: $(7 + 0.5) imes 2 = 15$. Thus, the person started with 15 kg of sugar.