QBM011
M121.
M122. Twenty-seven small red cubes are connected together to make a larger cube that measures 3 x 3 x 3. All of its external faces are painted white and the cube is dismantled.
M123. Find the remainder of 12345678987654321 divided by 328 ?
M124. In order to win a game of darts, a player must finish on exactly zero and their last dart must land in a double or hit the bull's eye.For example, if a player hit treble twenty (60 points), double twenty (40 points) and the bull's eye (50 points) they would score 150 points and could use this combination to finish.How many ways can a player finish from 150 points with three darts?
M125. In How many ways can a number 6084 be written as a product of two different factors?
M126. The 7th digit of (202)3 is
M127. Can you prove that (2n)! is divisible by 22n − 1?
M128. Sum of all two digit numbers which are divisible by 9?
M129. If the number 3402 is converted from base 10 to base x, it becomes 12630. what is the value of x ?
M130. How many of the numbers under 100 that are divisible by 10 can you make by adding four consecutive integers?
M131. Two railway stations, P and Q, are 279 miles apart. A train departs from P at 2pm and travels at a constant speed of 51 mph towards Q.
At 3pm a second train begins a journey from Q towards P at a constant speed of 60 mph.How far apart are the two trains twenty minutes before they pass each other?
M132. Can you prove that the expression 21n − 5n + 8n is divisible by 24?
M133. Divide 136 into 2 parts such that when one divided by 8 leaves the remainder of 3 and other divided by 5 it leave the remainder of 2. How many such sets are there
a. 3 b. 2 c. 4 d.1
M134. abc is a three-digit natural number so that abc = a! + b! + c!. what is the value of (b + c)a ?
M135. Prove that for any number that is not a multiple of seven, then its cube will be one more or one less than a multiple of 7.
END
Solution please
Any one has solved Question M126.
QBM.126
Q. Find the 7th digit of 202^3 Ans. 202^3= (200+3)^3 as 200 >> 3 so in the 7th digit there will be no effect of 3s power. 200^3=8000000 7th digit is 8. If nt undrstud thn email me at dip.smart@rediffmail.com Thanking You, Dipanjan
HUMRAJ!!!!
M122. Twenty-seven small red cubes are connected together to make a larger cube that measures 3 x 3 x 3. All of its external faces are painted white and the cube is dismantled.
Sol.
no. of cubes which are no face colour = 1
no. of cubes which are one face colour = 12
no. of cubes which are two face colour = 6
no. of cubes which are three face colour = 8
HUMRAJ!!!!!!!!
M125. In How many ways can a number 6084 be written as a product of two different factors?
Sol.
6084 = 22 x 32 x 132
the number of ways can a number 6084 be written as a product of two different factors
= (1/2){(3)(3)(3)+1}
= 14
HUMRAJ!!!!!!
M128. Sum of all two digit numbers which are divisible by 9?
Sol.
= 18 +27 +36................................. +99
= 9(2 +3 +4.....................................+11)
= 9[{(11x12)/2}-1]
= 585
HUMRAJ!!!!!!
M129. If the number 3402 is converted from base 10 to base x, it becomes 12630. what is the value of x ?
Sol.
(3402)10 = (12630)x
3402 = 1.x4 + 2.x3 + 6.x2 + 3.x1 + 0.x0
3402 = x(x3 + 2.x2 + 6.x1 + 3)
hit and trail;
then x =7
HUMRAJ!!!!!!!!
M131. Two railway stations, P and Q, are 279 miles apart. A train departs from P at 2pm and travels at a constant speed of 51 mph towards Q.
At 3pm a second train begins a journey from Q towards P at a constant speed of 60 mph.How far apart are the two trains twenty minutes before they pass each other?
Sol.
the relative speed = 111 mph.
distance cover in 20 minutes = (2/3)(111) = 74 miles
HUMRAJ!!!!!!!!!
M133. Divide 136 into 2 parts such that when one divided by 8 leaves the remainder of 3 and other divided by 5 it leave the remainder of 2. How many such sets are there
a. 3 b. 2 c. 4 d.1
let two parts are x and y.
x = 8m + 3 and y = 5n + 2
x +y = 136
8m + 5n + 5 = 136
n = (131 - 8m)/5
n is multiple of 5.
then m = 2, 7, 12
only 3 possible sets.
QBM011
M122. Twenty-seven small red cubes are connected together to make a larger cube that measures 3 x 3 x 3. All of its external faces are painted white and the cube is dismantled.
Ans. The diffiernt cube will form set of {8 cubes with three faces painted white,12 cubes with two white faces, 6 cubes with one white,1 completely non painted cube}
The number of ways of arranging back it.
Type -1 are the edge cube of the bigger cube - No. of ways = orientation x places = 3^8 x 8!
Type-2 are the cube are inbetween the corner cubes and = 2^8 x 12!
Type-3 are the cube in the centre of the faces = 4^6 x 6!
Type -4 is orientation = 24 ways i.e. 4 each for one face
The net solution for number of ways of arranging back the cube back in original form = 3^8 x 8!x2^8x12!x4^6x6!x24 Ans.
And all possible rearrangements = 24^27x27! Ans.
M123. Find the remainder of 12345678987654321 divided by 328 ?
Ans. Here it cane be seen that the number isN = (111111..9 times )^2 is to be divided by 328 = 2^3 x 41
Now 111...9 times = 1 mod 2 thus N = 1 mod(2)
and 111..9 times = 4 mod (41) thus N = 16 mod (41)
The number which leaves 41+16 = 57 and it leaves remainder 1 by 2 thus remainder is 57 Ans.
M125. In How many ways can a number 6084 be written as a product of two different factors?
Ans. Its factors = 2^2x3^2x13^2 only one pair is same because it is perfect square. Thus all pairs -1
3x3x3 -1 = 26 ways. and balance ways are same i.e. the number changes position i.e X x Y and Y x X thus total pairs 26/2 = 13 ans.
M126. The 7th digit of (202)^3 is
Ans. The number is 8(101)^3 = 8x(100+1)^3 = 8(1000000+300(101)+1)=8242408 it is 8. From visible inspection of eq. also it can be identified.
M128. Sum of all two digit numbers which are divisible by 9?
Ans. 18+...+99 Here 99 = 18+(n-1)9 thus 10 terms ans sum = 5x(18+99) =585 Ans
M131. Two railway stations, P and Q, are 279 miles apart. A train departs from P at 2pm and travels at a constant speed of 51 mph towards Q.
At 3pm a second train begins a journey from Q towards P at a constant speed of 60 mph.How far apart are the two trains twenty minutes before they pass each other?
Ans. The the relative speed = 51+60 =111 mph then they are 111/3 miles apart before twenty minutes.
M133. Divide 136 into 2 parts such that when one divided by 8 leaves the remainder of 3 and other divided by 5 it leave the remainder of 2. How many such sets are there
a. 3 b. 2 c. 4 d.1
Ans. Let the two parts be x and y then x+y =136 and x = 8k+3 and y = 5l+2
Then 8k+3 + 5l+2 = 136 and 8k+5l =131
The integeral solution to be considered. Thus values of (k,l) = { (2,23),(7,15),(12,7) } Thus three possible combinations.
answer to M134 is 9
solution
answer to M134 is 9
solution abc is natural number =a!+b!+c!
taking a as 1 b as 4 and c as 5 we get
natural number as 145 and 1!=1 +4!=24+5!=120
sum also comes out as 145
therefore (b=c)^a
(4+5)^1
hence answer is 9