Number Patterns and Math Puzzles

Son's age = 2. Father's age = 11 × 2 = 22. Grandfather's age = 2 × 22 = 44. The grandfather is 44 years old. Option A (2) is the son's age, B (22) is the father's age, and D (88) would require 4× multiplier.

A prime number is divisible only by 1 and itself. 18 is divisible by 1, 2, 3, 6, 9, and 18, so it has multiple divisors and is not prime.

19 is a prime number because its only divisors are 1 and 19. It cannot be divided evenly by any other integer, making it a true prime.

This sequence lists prime numbers in ascending order - numbers divisible only by 1 and themselves. Starting with 2 (the only even prime), the series continues 3, 5, 7, 11, 13, 17, 19, and the next prime is 23. Options A (21 = 3x7), B (22 = 2x11), and D (24 = 2x12) are all composite numbers.

Using the formula Speed = Distance/Time: 200 kilometers ÷ 2 hours = 100 km/h. This is a direct application of the speed formula where distance traveled is divided by time taken. Option A (50) would result from dividing time BY distance incorrectly, B (400) from multiplying, and C (200) from just restating the distance.

This is an arithmetic progression where each term increases by 5: 5, 10, 15, 20, 25, so the next term is 30. The pattern is consistently adding 5 to each number (5+5=10, 10+5=15, etc.). Option A (40) would be skipping by 10s after 20, B (35) breaks the pattern, and D (25) repeats the previous term.

To convert weeks to minutes: 60 minutes/hour × 24 hours/day × 7 days/week = 10080 minutes/week. This is a straightforward chain conversion through the standard time units. Option A (14400) would be 10 days, and option B (1440) is only one day.

0.01 (1/100) is greater than 0.001 (1/1000). The more decimal places after the point, the smaller the number becomes. This is a basic place value concept in decimal numbers.

When comparing fractions with the same denominator, the fraction with the larger numerator is greater. Since 4 > 3, 4/5 > 3/5. You can also convert to decimals: 0.8 > 0.6.

Adding decimals with place values: 0.1 + 0.01 + 0.001 + 0.0001 = 0.1111. Line up by decimal point and add each column from right to left. This tests understanding of decimal place value.

This is a geometric sequence where each term is doubled to get the next. The pattern is: 2, 4, 8, 16, 32, 64, so the next term is 64 × 2 = 128. Each term equals 2^n where n is the position.

The camera schedule starts at 9:30, then every 30 minutes: 10:00, 10:30, 11:00... Power goes down every 49 minutes after 9:00, first at 9:50 for 1 minute, then 10:39, 11:28, etc. At 10:00 (camera on), power is ON (from 9:51 to 10:38). At 10:30 (camera on), power is ON. At 10:39 (power off), camera would be OFF. At 11:00 (camera on), power is ON (from 10:40 to 11:27). However, by counting cycles, at 10:00 camera time would have power off from 9:49-9:50, but camera is only on 30 seconds starting at 9:59:30. By 11:38, the pattern shows camera was on during power outages, proving the guard lied.

For natural numbers a > b > 0, the difference of squares is a² - b² = 45, which factors as (a-b)(a+b) = 45. The factor pairs of 45 are (1,45), (3,15), and (5,9). Each pair yields a valid solution for (a,b): (23,22), (9,6), and (7,2).

Byomkesh starts 358 m behind the thief. His speed is 10 m/s, thief's is 8 m/s, so relative speed is 2 m/s. Time to catch = 358/2 = 179 seconds = 2 min 59 sec. He catches the thief before his speed drops.

Initially, Byomkesh's relative speed is 10-9 = 1 m/s. In 3 minutes, he closes 180 m of the 358 m gap, leaving 178 m. After 3 minutes, his speed drops to 7 m/s while the thief continues at 9 m/s, making the relative speed -2 m/s (Byomkesh falls behind). Thus, he never catches the thief.

22! (22 factorial) = 22 × 21 × 20 × ... × 1 is an extremely large number that far exceeds 22, 32, or 20. Factorials grow faster than exponential functions, so 22! is the maximum possible number that can be formed using exactly two 2's with mathematical notation. 22! is approximately 1.124 × 10^21.

If a number leaves remainder 3 when divided by 17 and 23, then that number minus 3 is divisible by both 17 and 23. Since 3 itself satisfies this condition (3-3=0, which is divisible by any number), 3 is the least positive integer with this property. The LCM of 17 and 23 is 391, so other solutions are 394, 785, etc.

This sequence is 1, 9, 57, 345, ?. To find the pattern, examine the differences between consecutive terms: 9-1=8, 57-9=48, 345-57=288. These differences (8, 48, 288) follow a pattern where each difference is 6 times the previous difference: 8×6=48, 48×6=288. Therefore, the next difference is 288×6=1728, making the next term 345+1728=2073. Option D is correct.

The sequence 2, 3, 5, 7, 10, ? requires finding the pattern in differences: 3-2=1, 5-3=2, 7-5=2, 10-7=3. The differences are 1, 2, 2, 3. This follows a Fibonacci-like pattern where each new difference (after the first two) is the sum of the two previous differences: 1+2=3, and 2+3=5. Therefore, the next difference is 5, giving 10+5=15. Option A is correct.

The word INTERNATIONAL has 13 letters. Counting repetitions: I appears 2 times, N appears 3 times, T appears 2 times, A appears 2 times, and E, R, O, L each appear once. The number of distinct arrangements is 13! divided by the factorials of the repetition counts: 13!/(2! × 3! × 2! × 2!) = 6,227,020,800/48 = 129,729,600. Option D is correct. Options A and B ignore repeated letters entirely, while C is irrelevant.