PROBLEMS ABOUT ALGEBRA AND ELEMENTARY FUNCTIONS
INTEGERS
PROBLEMS ABOUT INTEGERS:
14. An even power of a natural number is a four-digit number, whose thousands digit
is 3 and units digit is 5. Find this number.
1.
Find the two-digit number, which is equal to sum of the tens digit and the square
of units digit.
2.
Prove that n  , x   st nx  1 is a composite number.
15. Find the four-digit number that satisfies the equality abcd  dcba  360 and
whose last three digits form an arithmetic progression.
3.
The units digit of a perfect square number is 5. Prove that the hundreds digit of
this number is even.
16. Find a three-digit perfect square number, whose product of digits is one less than
the square root of this number.
4.
n  , st n  2, prove that the units digit of 2
5.
Find the three-digit number, whose product of the factorials of its digits times 3/2
equals itself.
18. Prove that n   , 9  1 may end with one zero, at most.
6.
Find a four-digit perfect square number, whose thousands digit is equal to tens
digit and hundreds digit is one more than units digit.
19. Find last two digits of 7
7.
Find a four-digit perfect square number, whose units and hundreds digits are equal
to tens and thousands digits, respectively.
8.
......
......
If A, B and C are three natural numbers, st A  111



1 , B  111



1 and
20. Prove that, if the expression ax  bx  cx  d is equal to 1 and 2, for
x  19 and x  62, respectively, then the coefficients a, b, c and d can not be
integers.
2
n
17. Prove that   111
......
.....



1  222.


2 is a perfect square. Also find
 1 is 7.
2 m digits
m1 digits
2 m digits
C  666
......


6 , then prove that A  B  C  8 is a perfect square number.

.
m digits
n
9
9
9
.
3
2
21. Prove that, n, k  , the number   2  n
the product of consecutive natural numbers.
3k
k
 4  n  10 cannot be written as
m digits
9.
Prove that every even number, which is not divisible by 4, can not be written as
2
2
(a  b ), a , b  Z .
10. Find three-digit number abc that satisfy the equality abc 
bca  cab
2
.
22. Find all natural numbers x, for which the expression 22  x  5 is the square of a
natural number.
23. Prove that, of all three-digit numbers, only number 145 is equal to sum of factorial
of its digits.
2
24. Find the four-digit number abca, which is equal to (5  c  1) .
2
11. Prove that, x  , the function y  ax  bx  c takes integer values, if and only
if 2a, a  b and c are integers.
2
12. Let p be a prime number. Prove that 8 p  1 is composite for all p  3.
13. Prove that n  , 21n  4 and 14n  3 are relatively prime.
25. Prove that there are 25 two-digit numbers that satisfy the following property:
When the order of the numbers is reversed, the number decreases by the square of
the number.
26. A six-digit perfect square number is divided into two three-digit numbers, by
taking the first three digits as one number and the last three digits as the other. If
1
PROBLEMS ABOUT ALGEBRA AND ELEMENTARY FUNCTIONS
INTEGERS
one of the two numbers is eight times the other, and the difference between them
is a perfect square, find this six-digit number.
27. Find the integer values of a, if the polynomial ( x  a)  ( x  10)  1 can be written
as the product of two polynomials ( x  b)  ( x  c), where b and c are integers.
DIVISIBILITY:
39. Find the remainder, when the number 520 is divided by 24.
40. Prove that, n   , the number 37
5
28. The difference of the cubes of two natural numbers, that are divisible by 4, is a
four-digit number, divisible by 91. Find these numbers.
41. Prove that, the number (3299  6)
105
29. The G.C.D of the two three-digit numbers m and n is 6 and L.C.M is a six-digit
m
number. Find the sign of the value tan (
).
350
105
18
 16
n 1
n
 23 is divisible by 7.
 1 is divisible by 112.
is divisible by 181.
43. Prove that, if k , m, n   , the number 5
5 k 1
4
5 m 2
2 n 2
46. Prove that, n  , the number 72
p
47. Find the remainder, when the number (9674  28)
 28
6
33. Prove that, if 2  1, where p is a prime number, is prime, then the sum of all
p1
p
divisors, except itself, of   2
 (2  1) is equal to N.
k
34. Prove that the product of all positive divisors of natural number n is equal to n 2 ,
where k is the number of positive divisors of n.
35. Given that prime number p is the difference of squares of two natural numbers.
Find these numbers.
36. Prove that, if the digits of a three-digit number are in an arithmetic progression,
then this number is not a prime number.
n !( n  1)!
 47
2n
n
32. Prove that, if 2  1 is a prime number, then n is a power of 2.
( 2n)!
5n
45. Prove that, a, b, n   and n  0, the number (7a  3)
divisible by 7.
31. Prove that the sum of consecutive odd numbers is composite.
37. In order to obtain a three-digit number, with same digits, what must be the last
term of the summation 1 2  3  4..............n.
3
is divisible by 11.
44. When the natural number N is divided by 3 and 37, the remainders are 1 and 33,
respectively. Find the remainder, when N is divided by 111.
30. Find all values of natural numbers m and n that satisfy the following two
n
2
2
conditions: a) m  9 n is a prime number, b) 2 lg  1  0 .
m
38. Prove that
4
42. Prove that, the number 3
n2
48. Prove that, the number 2
60
7
30
n
n
50. Prove that, n  , ( a
a, b   and a  b  0.
2 n1
b
51. Prove that, n  , (a
a, b   and a  b  0.
2n
2 n1
2 n1
15
 (7b  25)
2 n1
is
is divisible by 25.
is divided by 39.
is divisible by 13.
49. Prove that, n  , (a  b ) is divisible by (a  b) , where a and b are distinct
natural numbers.
b
2 n1
2n
) is divisible by (a  b) , where
) is divisible by (a  b) , where
n
52. Prove that, n  , (4  15n  1) is divisible by 9.
is a natural number.
2
PROBLEMS ABOUT ALGEBRA AND ELEMENTARY FUNCTIONS
INTEGERS
53. Prove that, for all odd integers a i , not divisible by 3, where i  1,2,3,.......,2n ,
the number
2
a1
2
 a2
2
2
 a 3 ....... a 2 n 1
2
 a 2n
2
is divisible by 24.
n
54. Prove that, n, k   and ( k  1), the number 3
k
7
55. Prove that, if the sum of x and y is divisible by 7, then the sum x  y is divisible
by 49.
2
2
2
57. Find the integer values of x, such that the number a  ( x  a  x  a  1) is
divisible by 6, for any integer a.
58. Prove that n  , the number 5
2 n1
2
n1
3
n1
2
2 n1
60. Prove that n, a  , the number (a
5
is divisible by 19.
73. Prove that n  , the number 2
divisible by 31.
61. Prove that the indivisibility of integer n by 2 and 3 is necessary and sufficient for
2
the divisibility of 4 n  3n  5 by 6.
2
62. Find all natural numbers n, such that n  1 is divisible by n 1.
2( n 1)
5
2n
5
5
72. Prove that n  , the number 7
 a ) is divisible by 30.
63. Prove that, for all nonnegative integers n, the number 3
divisible by 117.
69. Prove that, if a and b are relatively prime natural numbers, then the numbers
2
2
a  b and a  b do not have a common divisor, greater than 2.
71. Prove that, ( x  y )  ( y  z )  ( z  x ) is divisible by 5 ( x  y )  ( y  z)  ( z  x ) ,
where x, y and z are integers, that are not equal in pairs.
59. Prove that when a prime number is divided by 30, the remainder is never
composite, i.e. this remainder equals either 1 or it's a prime number.
4 n1
2n
5n-1
n
3
3n  2
2
2n
is
65. Prove that, the sum of the squares of two integers a and b is divisible by 7, only
when each of the numbers a and b is divisible by 7.
4
2
n
 30  21 is divisible by 16.
5n 2
2
n
5n 3
n
3
2
.........2  2  1 is
n1
n
 1)  5  (5
n1
n
 5 ) is
2
3
2
76. Prove that n  , the number 2n  3n  n is divisible by 6.
n
2
2n
74. Prove that n  , the number 16  (2  1)  9  (9
divisible by 7.
4
2
3
3
75. Prove that n  , the number n  2n  11n  62n is divisible by 24.
64. Prove that x  , the number x  3x  5 is not divisible by 121.
2
 1 is divisible by n, where n, k   .
70. When a 6-digit number, consisting of same digits, is divided by a 4-digit number,
also consisting of same digits, the quotient is 233 and a certain remainder is
obtained. After taking off one digit from the dividend and divisor, the quotient
does not change, but the remainder decreases by 1000. Find the dividend and
divisor.
56. Prove that, among 39 consecutive natural numbers, necessarily, there is such a
number whose sum of digits is divisible by 11.
3
k 1
68. Prove that, the number 70a  21b  15c  N is divisible by 105, if, when the
integer N is divided by 3,5 and 7, the remainders are a, b and c, respectively.
 1 is not divisible by 5.
7
k
67. Prove that, the number ( n  1)  ( n  1)
n
n
n
77. Prove that n  , the number 30  4  (3  2 )  1 is divisible by 11.
78.
2
66. Prove that x , y   , the number ( x  y  4 x  y )  ( x  x  2) is divisible by
216.
3