QBM008
M76. How many zeroes will be there at the end of 12 x 22 x 32 x . . . . . . x 1002 ? M77. In my pocket I have exactly 15p, comprising of eight coins made up of 1p, 2p and 5p pieces. How many of each coin do I have? M78. When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12? M79. If every combination of the digits 1,2,3,4 was written down, what would be the sum of the numbers formed? M80. Find the sum of all even four digit numbers that are divisible by first five natural numbers in number system? M81. If F(x)= sum of all the digits of x, where x is a natural number, then what is the value of F(101) + F(102) + F(103). . . . + F(200) M82. A box contains a mixture of black and white discs, with more black than white discs. A game is played by taking one disc at a time, at random, and without replacement. If an equal number of each colour have been removed the game stops and the player wins. It is found that the player has an equal chance of winning or losing. If the box contains twelve discs in total, find the number of black discs M83. If F(x)= sum of all the digits of x, where x is a natural number, then what is the value of F2(101) + F2(102) + F2(103). . . . + F2(110) M84. During a football match in a tournament a goalkeepers save rate was given as 33%. After saving the next shot on target it rose to 40%. How many more shots on target does he need to save to raise his save rate to 50%? M85. The number of pairs of consecutive positive integers whose product is less than 2000 is? What is the highest power of 31 in 1000!? M86. A train, length 1/2 mile, travelling at a constant speed of 30 miles per hour enters a tunnel which is 2 miles long. How long will it take for the train to completely pass through the tunnel? M87.How many numbers in the first 100 natural numbers are not divisible by 2, 3 or 5. M88. When a number is divided by 29, it leaves a remainder of 19. What will be the remainder when the number is divided by 12? M89. A train travels a distance of 90 miles from A to B in one hour. Another train sets off at the same time and travels from B to A, taking two hours to complete the journey.How many miles from A did the two trains cross? M90. How many zeros will be there at the end of 37! ? END
QBM008
M76. How many zeroes will be there at the end of 1^2 x 2^2 x 3^2 x . . . . . . x 100^2 ?
Ans It is 100!x100! thus the power of 5 in 100! = 20+4 =24 thus 100!x100! = 48 Ans
M77. In my pocket I have exactly 15p, comprising of eight coins made up of 1p, 2p and 5p pieces. How many of each coin do I have?
Ans. x+y+z = 8 and x+2y+5z = 15 Here z can be 1 or 2
Let z is 2 then x+y=6 and x+2y = 5 as x > 0 and y > 0
which indicates z =1
the equation become x+y=7 and x+2y=10 which gives y=3 and and x =4
Thus 1p coin =4 ;2p coin =3 and 5p coin are 1 no.
M78. When a number is divided by 36, it leaves a remainder of 19. What will be the remainder when the number is divided by 12?
Ans. N = 36K + 19 thn remainder when divided by 12 is 7 Ans.
M79. If every combination of the digits 1,2,3,4 was written down, what would be the sum of the numbers formed?
Ans. Total combination is 4! (i.e wo repaetion) = 24 Nos and the sum is 6x(1111)(1+2+3+4) =60x1111=66660
M80. Find the sum of all even four digit numbers that are divisible by first five natural numbers in number system?
Ans. First five natural no. 1,2,3,4,5 LCM = 60 thus
First even digit is 1020 and last is 9960 and no of terms = 150 Nos thus sum = 823500
M81. If F(x)= sum of all the digits of x, where x is a natural number, then what is the value of
F(101) + F(102) + F(103). . . . + F(200)
Ans. Total no is 100
1 added 199 time = 199 in thosusand place 2 =201
and sum of digits in unit is 100x(1+2+3+4+...+9)=4500
and sum of no in ten place is 10x(1+...) = 450 The total is 4950+201 =5151
M82. A box contains a mixture of black and white discs, with more black than white discs. A game is played by taking one disc at a time, at random, and without replacement. If an equal number of each colour have been removed the game stops and the player wins. It is found that the player has an equal chance of winning or losing. If the box contains twelve discs in total, find the number of black discs
Ans. The solution may be (7,5)
M84. During a football match in a tournament a goalkeepers save rate was given as 33%. After saving the next shot on target it rose to 40%. How many more shots on target does he need to save to raise his save rate to 50%?
Ans. x/y = 33/100 and x+1/y+1 =40/100 then x+z/y+z =50/100 what is z ? % variation is high thus the value of x,y,z is small numbers.
Let 3/9 then 4/10 and thus 3+z/9+z =1/2 thus 6+2Z = 9+z or z = 3 goals.
M85. The number of pairs of consecutive positive integers whose product is less than 2000 is?
What is the highest power of 31 in 1000!?
Ans. The a(a^2-1) < 2000 This gives maximum value of a = 12 thus the no. of pairs =11
The highest power of 31 in 1000! = 1000/3+1000/9+1000/27+1000/81+1000/243+1000/729 =333+111+37+12+4+1=510 Ans.
M86. A train, length 1/2 mile, travelling at a constant speed of 30 miles per hour enters a tunnel which is 2 miles long. How long will it take for the train to completely pass through the tunnel?
Ans. The time take to completely pass through the tunnel = 2+1/2 =5/2 miles Now 30 miles in 60 mts or 1/2 miles in 1 mts Thus 5 mts is answer.
M87.How many numbers in the first 100 natural numbers are not divisible by 2, 3 or 5.
Ans. ~N( 5 or 3 or 2) = 100 - N( 5 or 3 or 2)
Now N( 5 or 3 or 2 ) = N(5)+N(3)+n(2) - N(5 and 3) - N( 3 and 2 ) -N(2,5) + N(5 and 3 and 2) By solving the above as A.P and finding the terms can be solved.
M89. A train travels a distance of 90 miles from A to B in one hour. Another train sets off at the same time and travels from B to A, taking two hours to complete the journey.How many miles from A did the two trains cross?
Ans. Relative velocity = 90/1+90/2=135 mile/hr the time taken to meet 1 hr = 135 miles the 90 miles => 90/135 hr Then distance covered 90x90/135 miles = 8100/135 from A = 60 miles
M90. How many zeros will be there at the end of 37! ?
Ans. The no of zeros = 37/5+37/25 = 7+1 = 8 Ans.