The sum of three consecutive odd numbers is 93. What is the middle number?

6² -1 = 35, which is divisible by both 5 and 7.

LCM of 16, 18 and 20 = 720
? Required number = 720k + 4
Where, k is a natural number to be divisible by 7, (720 k + 4) will be a multiple of 7.
Smallest value of k = 4
? Required number = 720 x 4 + 4 = 2884

If A Score 100 then B scores 80 and C score 72.
? B scores 80 while C scores 72
? B scores 100 while C scores 72 * 100 / 80 = 90 points
Hence, in 100 points B can give (100 ? 90) = 10 points

Sum of the ages of 4 men five years ago
= 48 x 4
=192 yr

And sum of present ages of the four men
= 192 + 5 x 4
= 212 yr

Now ( 212 + Age of the new man ) / 5 = 50
? Age of the new man = 50 x 5 - 212
= 38 yr.

( 2³ - 2) = 6 is the largest natural number that divides ( a³ - a) for every number a.

We know that, a number is divisible by 11 when the sum of the difference between the sum of its digit at even places and sum of its digit at odd places is either 0 or the difference is divisible by 11.

For 93455120
Sum of even places = 3 + 5 + 1 + 0 = 9
Sum of odd places = 9 + 4 + 5 + 2 = 20

There difference = 20 - 9 = 11
which is divisible by 11 so 93455120 is divisible by 11.

Number between 0 and 11 which are multiple of 2 or 3
= 11/2 + 11/3 - 11/6 = 5 + 3 - 1
= 7

Number between 0 and -11 which are multiple of 2 or 3
= 11/2 + 11/3 - 11/6 = 5 + 3 - 1
= 7

? Number be 15, including 0

Let the number be x.
44² < x < 45²
? 1936 < x < 2025 ...(i)

From equation (i), the required number will be any number between 1936 and 2025. Since one part of the number is the square of 6 means one factor is 36.

? L, C, M of 36 and 5 = 180
? Number will be multiple of 180 i,e.,
180 x 11 = 1980 the only value which satisfies the equation (i).

If A run 1000 meters then B runs 900 meters and C runs 800 meters.
? B runs 1350 meters while C will run 800 * 1350/ 900 = 1200 meters

Hence, B can beat C by (1350 ? 1200) = 150 m

S = {HH, HT, TT, TH}
and E = {HH, HT, TH }
? P(E) = n(E)/n(S) = 3/4