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?
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.
+128474 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
+128474 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
+128474 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
