1. Q. Show that there is no positive integer n so that √ (n-1 )+ √(n+1) is rational
2. Q. Prove that one of every three consecutive positive integers is divisible by 3 ?
3. Q. Show that m3-m is divisible by 6 for each natural number m.
4. Q. What is the product of a non-zero rational and an irrational number?
5. Q. Find the least number that is divisible by all the numbers from 1 to 10 ?
6. Q. Prove that n2-n is divisible by 2 for every positive integer n?
7. Q. Show that
7. Q. Show that
(a) The sum, the difference, and the product of two even numbers is always even.
(b) The sum and the difference of two odd numbers is always even, whereas their product is always odd.
8. Q. Prove that the expression y ( y + 1) always represents an even number, where y is any positive integer.
9.Q. Prove that every positive integer is of the form 3 p , 3 p + 1, or 3 p + 2, where p is any integer?
10. Q. if 'a ' and 'b ' are odd positive integers,a2+b2 is even Prove that it is not divisible by 4?
11.Q. Show that the expressions given below are composite numbers.
8. Q. Prove that the expression y ( y + 1) always represents an even number, where y is any positive integer.
9.Q. Prove that every positive integer is of the form 3 p , 3 p + 1, or 3 p + 2, where p is any integer?
10. Q. if 'a ' and 'b ' are odd positive integers,a2+b2 is even Prove that it is not divisible by 4?
11.Q. Show that the expressions given below are composite numbers.
12.
Q. Find the largest number which divides 75 and 130, leaving remainders 3 and 5, respectively?
(a) 3 × 5 × 7 × 23 + 2 × 7 × 11 × 13 (b) 29 × 35 + 14 (c) 3power 4 + 6power 3
(a) 3 × 5 × 7 × 23 + 2 × 7 × 11 × 13 (b) 29 × 35 + 14 (c) 3power 4 + 6power 3
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