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1) What is the smallest positive integer n such that the rightmost three digits of n! and (n+1)! are the same?

2)N=1991*1993*1995*1997*1999. What is the sum of the hundreds, tens and units digits of N?

3) What is the remainder when (17^77) is divided by 35?

4) Find the smallest positive multiple of 21 that has no digit larger than 1.

5) Four positive integers A, B, C, D and  have a sum of 36. If A+2=B-2=C*2=D/2, what is the value of the product of A*B*C*D?

6) There is a rectangular patio. If we increase both the length and width by 2 feet, the area of the patio will increase by 38 square feet. If we increase the length by 2 feet and decrease the width by 2 feet, the area of the patio will decrease by 2 square feet. What is the area of the patio?

7) Suppose b and c are positive integers. When b^2 is written in base c, the result is 121_c. When c^2 is written in base b, the result is 71_b. What is b+c?

 Feb 5, 2019
edited by Guest  Feb 5, 2019
 #1
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+1

1) - The smallest positive n! and (n + 1)! are: 10! =3,628,800  and  11! =39,916,800

 

2) 1991*1993*1995*1997*1999 = 31601836203377055

     0 + 5 + 5 = 10

 

3) 3) What is the remainder when (17^77) is divided by 35?

     17^77 mod 35 = 12

 

4)  Find the smallest positive multiple of 21 that has no digit larger than 1.

     481 x 21 =10,101.

 Feb 6, 2019
 #2
avatar+98129 
+1

5) Four positive integers A, B, C, D and  have a sum of 36. If A+2=B-2=C*2=D/2, what is the value of the product of A*B*C*D?

 

A+ 2 = B - 2    ⇒  A + 4 = B

 

and

 

A + 2 = C*2

 

[ A + 2 ] / 2 = C

 

and

 

A + 2 = D /2

 

2[ A + 2] = D

 

So     ... A + B + C + D   = 36            substituting

 

A + (A + 4) + [ A + 2] / 2 + 2[ A + 2] = 36      simplify

 

4.5A + 9 =  36

 

4.5A = 27

 

A = 6     B = 10    C = 4     D = 16

 

So

 

ABCD  =  3840

 

 

cool cool cool

 Feb 6, 2019
 #3
avatar+98129 
+1

6) There is a rectangular patio. If we increase both the length and width by 2 feet, the area of the patio will increase by 38 square feet. If we increase the length by 2 feet and decrease the width by 2 feet, the area of the patio will decrease by 2 square feet. What is the area of the patio?

 

Call the imensions of the patio L and W....so the area = LW

 

We know that

 

(L + 2) (W + 2) = LW + 38

LW + 2W + 2L + 4 = LW + 38

2(W + L) = 34

W + L = 17

L = 17 - W   (1)

 

And we also know that

 

(L + 2) (W - 2) = LW - 2

LW + 2W - 2L - 4 = LW - 2

2W - 2L =  2

W - L = 1

sub in (1) for L

W  - (17 - W) = 1

2W - 17 = 1

2W = 18

W = 9

 

And L = 17 - 9  =  8

 

So....the area is  LW = (8)(9) = 72 ft^2

 

 

cool cool cool

 Feb 6, 2019
 #4
avatar+98129 
+1

7) Suppose b and c are positive integers. When b^2 is written in base c, the result is 121_c. When c^2 is written in base b, the result is 71_b. What is b+c?

 

121c  =  c^2 + 2c + 1   =  b^2      (1)

 

c^2  =  7b + 1  

 

c^2 - 1 = 7b

 

[ c^2 - 1 ] / 7 = b    ⇒   [c^2 - 1 ]^2  / 49 = b^2      (2)

 

Sub (2) into (1)

 

c^2 + 2c + 1  = [ c^2 - 1 ]^2 /49

 

49(c + 1)^2 = [ (c + 1) (c - 1) ] ^2 

 

49 =  (c - 1)^2

 

So....c = 8

 

And b =  [ c^2 - 1 ] / 7   =   [ 64 - 1 ] / 7 =  9

 

So

 

b + c  =    9 + 8   =   17

 

 

cool cool cool

 Feb 6, 2019

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