Math Puzzles and Number Series

A number is divisible by 3 if the sum of its digits is divisible by 3. For the sum to be divisible by 3, the total digit sum including the unknowns (*) must be a multiple of 3. Testing the options, 5 is the maximum value that satisfies the condition.

If A is even, A^k ends in 6 if k is a multiple of 4 (except for A ending in 0). If A ends in 0, A^k ends in 0. Since 4n-4 is a multiple of 4 for n>1, the unit digit must be 0 or 6.

Factor out 3^71: 3^71(1 + 3 + 9) = 3^71 × 13. Since 13 divides the expression, and 39 = 3 × 13, the expression is divisible by 39. Check: 34, 26, and 19 are not factors of 13 or 3.

The Maclaurin series for cos(x) contains only even powers of x with alternating signs, starting with 1. Option A is the series for sin(x) (odd powers only), Option C is e^x (all positive terms), and Option D is a geometric series.

The Taylor series expansion for cos(x) is 1 - (x^2/2!) + (x^4/4!) - .... It contains only even powers of x and alternating signs. Option 103095 is sin(x), 103097 is exp(x), and 103098 is the expansion for 1/(1+x).

The number must contain 11^2 and 3^3. The current expression is a * (2^2)^3 * (2*3)^2 * 13^11 = a * 2^8 * 3^2 * 13^11. To have 11^2 and 3^3 as factors, 'a' must provide 11^2 and one more 3 (since we have 3^2). Thus, a = 121 * 3 = 363.

The LCM of 7, 10, 15, 21, and 28 is 420. To find the greatest five-digit number, divide 99999 by 420 (remainder 39). 99999 - 39 = 99960. This number is exactly divisible by all the given factors, unlike the other options.

10^5 is 100,000. 100,000 - 7 = 99,993. A number is divisible by 3 if the sum of its digits is divisible by 3. 9+9+9+9+3 = 39, which is divisible by 3. It is odd, so not divisible by 2 or 8.

The sequence follows the pattern n^2 - 1 and n^2 + 1 alternately for specific integers: 2^2+1=5, 3^2-1=8, 5^2+1=26, 7^2-1=48, 11^2+1=122. The next prime is 13, so 13^2-1 = 168. 170 (13^2+1) is wrong because the subtraction/addition alternates.

The pattern is based on factorials or increasing multipliers: 2, 24 (212), 720 (2430). More accurately, these are 2!, 4!, 6!. The next term should be 8! = 40320. The other options are arbitrary numbers that do not fit the factorial growth pattern.

The sequence is composed of three interleaved patterns: the first is a constant '1' (1st, 4th, 7th terms); the second is 2, 4, 8 (powers of 2); the third is 3, 9, 27 (powers of 3). The 9th term follows 3 and 9, so 3^3 = 27.

This is a number sequence pattern. The sequence alternates between perfect squares: 1681 (41^2), 1764 (42^2), 1849 (43^2), and then 1936 (44^2). The pattern is: square(n), n, square(n+1), n+1, square(n+2), n+2, square(n+3). Following this pattern, after 43 comes 1936 which is 44^2.

The pattern is: multiply by 9 and subtract 8, then multiply by 8 and subtract 20, then multiply by 7 and subtract 14. The pattern uses decreasing multipliers (9, 8, 7) and subtracts values related to the multiplier: 4×9-8=28, 28×8-20=204, 204×7-14=1414, 1414×6-12=8472. However, based on the given sequence 4, 36, 292, 2034, the next term following the established pattern would be 18724.

The complete proverb is 'A man is known by the company he keeps,' meaning you can judge someone's character by the friends and associates they choose. Company here refers to the people one associates with.