17. Find the least positive integral value of z for which there exists different positive integers x, y and z that
satisfy x
2 + y
2
= z
3
.
18. There are three distinct positive integers whose sum is equal to the product of the largest two of them.
Find the product of these three integers.
19. A pair of ‘twin primes’ is a pair of primes that are consecutive odd integers. Find the largest integer that is
a divisor of the sum of the two elements in every pair of twin primes, if these two elements are each prime
greater than 3.
20. Find the integer x for which x
5 = 656356768.
21. In base 8, the four-digit numeral bbcc is the square of the two-digits numeral aa. Find the value of (a + b + c).
22. When 16 is subtracted from a three-digit number abc (greater than 100) and the resulting difference is divided
by 2, the result is a three-digit number (greater than 100) cba. If a + b + c = 20, find (2a + 3b + 4c).
23. Let a, b and c be positive integers such that ‘a’ is the cube of an integer, c = b + 1, and a
2 + b
2 = c
2
. Find
the sum of digits of least possible value of c.
24. Let N be the smallest natural number which becomes 9 times if a zero is inserted between its units digit
and tens digit. Find the sum of digits of N.
25. abc and cba are, respectively, the base nine and base seven numerals for the same positive integer. Find
the sum of digits of this integer when expressed in base ten.
26. Find the number of three-digit numbers for which one obtains, when dividing the number by 11, the sum of
the squares of the digits of the initial number.
Find the number of triplets of prime number (p, q, r) satisfying 3p
4 – 5q
4 – 4r
2 = 26.
q q q
Answers
Answer:
17. Find the least positive integral value of z for which there exists different positive integers x, y and z that
satisfy x
2 + y
2
= z
3
.
18. There are three distinct positive integers whose sum is equal to the product of the largest two of them.
Find the product of these three integers.
19. A pair of ‘twin primes’ is a pair of primes that are consecutive odd integers. Find the largest integer that is
a divisor of the sum of the two elements in every pair of twin primes, if these two elements are each prime
greater than 3.
20. Find the integer x for which x
5 = 656356768.
21. In base 8, the four-digit numeral bbcc is the square of the two-digits numeral aa. Find the value of (a + b + c).
22. When 16 is subtracted from a three-digit number abc (greater than 100) and the resulting difference is divided
by 2, the result is a three-digit number (greater than 100) cba. If a + b + c = 20, find (2a + 3b + 4c).
23. Let a, b and c be positive integers such that ‘a’ is the cube of an integer, c = b + 1, and a
2 + b
2 = c
2
. Find
the sum of digits of least possible value of c.
24. Let N be the smallest natural number which becomes 9 times if a zero is inserted between its units digit
and tens digit. Find the sum of digits of N.
25. abc and cba are, respectively, the base nine and base seven numerals for the same positive integer. Find
the sum of digits of this integer when expressed in base ten.
26. Find the number of three-digit numbers for which one obtains, when dividing the number by 11, the sum of
the squares of the digits of the initial number.
Find the number of triplets of prime number (p, q, r) satisfying 3p
4 – 5q
4 – 4r
2 = 26.
q q q