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QBM010

M106. For what value of 'n' will the remainder of 351n and 352n be the same when divided by 7? M107. If a is real then the minimum value of a2 - 8a + 11 is ________? M108. Prove that for any natural number, n, there exists a sequence of (n 1) consecutive numbers that are composite. M109. Convert 101001001 from base 2 to base 5 M110. Prove that 99n ends in 99 for odd n M111.When Christmas trees are planted they should stand at least 2 metres away from one another whilst growing.What is the maximum number of trees that can be planted in one square kilometre? M112. Convert 54321 from base 6 to base 10 M113. A wooden cube has each of its six faces painted either white or grey. How many different cubes can be made out of it? M114.96 + 1 when divided by 8, would leave a remainder M115. How many numbers below one hundred are divisible by both 2 and 3? M116. For what value of p , 2p + 3p can be a perfect square. Ans: never M117. Find the largest number which when divides 77, 147 and 252 leaves the same remainder a. 50 b. 35 c. 45 d. 30 M118. Prove that seven is the only prime number that is one less than a perfect cube. M119. What smallest number should multiply 34300, in order to make it a perfect square? M120. 3p + 3q = 2430 and it is known that p & q are positive then p + q equals a. 4 b. 10 c. 12 d. 8 END

QBM010

M106. For what value of 'n' will the remainder of 351^n and 352^n be the same when divided by 7?
Ans.  351 = 1 mod (7) and 352 = 2 mod 7 then 1^n = 2^n thus n=0 is the answer.

M107. If a is real then the minimum value of a2 - 8a + 11 is ________?
Ans.  Here it can be put as (a-4)^2 -5 thus minimum value is -5 Ans.

M110. Prove that 99n ends in 99 for odd n
Ans 99 will end in 01 for even values and end in 99 for odd values.
M111.When Christmas trees are planted they should stand at least 2 metres away from one another whilst growing.What is the maximum number of trees that can be planted in one square kilometre?
Ans. 1000 m thus 0 + 2+...+1000 thus no. of plant s = 501 and next row will have 499 plants at distance 2/(3)^0.5 Possible pairs = (1000/2)*1.732= 866
        Thus 2 rows = 501+499 =1000 plants thus total plants = 1000x433 = 433000 Ans.

M113. A wooden cube has each of its six faces painted either white or grey. How many different cubes can be made out of it?
Ans.  2^6 = 64 cubes

M114. 9^6 + 1 when divided by 8, would leave a remainder
Ans.  2 Ans

M115. How many numbers below one hundred are divisible by both 2 and 3?
Ans.   6 + 12+....+ 96 here Tn = 96=6+(n-1)6 or n =16 ans.

m106

for m106 the ans is 3

 

(351^n/)7=(1^n)/7

(352^n)/7=(2^n)/7

now,for n=3,remainder will be 1 in each case.therefore,ans is 3

ans. of no. 1

351/7=1 remainder, 352/7=2  now  1^n=2^n   n=3k   where k is any integar because  2^3=8 rem.=1 2^6=64 rem.=1

M119. What smallest number

M119. What smallest number should multiply 34300, in order to make it a perfect square?

Soln. As, 343100=7^3 x 10^2
       least no. is 7...

Giv ur best to d world,
Nd d best will cm back to u...

Regards,
Dipanjan......

M106 answer will be n=3 and

M106 answer will be n=3 and not 0 since (1^n)/7 and (2^n)/7 must have same remainder thus n=3

the answer to M106 would be

the answer to M106 would be n=3

and not  n=0

M120.

ans is 10;
proof : 3^p + 3^q = 2430
and 2430 = 3^4 x 10
therefore, 3^p + 3^q = 3^4 x 10
or 3^(p-4) + 3^(q-4) = 10

above implies that 3^(p-4) and 3^(q-4) are single digit, obtained as power of three and add up to 10.
single digit numbers that strike to fit into the above equation are 9 and 1
i.e 9+1 = 10
thus we get (p,q) = (6,4) or (4,6) in any case
p+q = 10.

hope things are clear.