Question:
1) Prove that if x and y are odd positive integers, then x square plus y square is even but not divisible by 4?
2)Prove that if a positive integer is of form 6q+5, then it is of form 3q+2 for some integer q, but not conversely?
3)A mason has to fit a bathroom with square marble tilesw of the largest possible size. The size of the bathroom is 10 ft. by 8 ft. What would be the size in inches of the tiles required that
has to be cut and how many such tiles are required?
4)Prove that root 6 is irrational?
5)Show that there is no positive integer n for which root n-1 plus root n+1 is rational?
6)Can two numbers have 16 as their HCF and 380 as their LCM? Give reason.
7)wht is the smallest number that, whaen divided by 35,56 and 91 leaves remainder of 7 in each case?
8)Find the least number that is divisible by all the numbers between 1 and 10 (both inclusive).
Answer:
1. Any odd positive integer is of the form 2q + 1, where q is some integer.
Let x = 2n + 1 and y = 2m + 1, where m and n are some integer.
Now, x2 + y2 = (2n + 1)2 + (2m + 1)2
=> x2 + y2 = 4n2 + 4n + 1 + 4m2 + 4m + 1
=> x2 + y2 = 4(n2 + m2 + n + m) + 2
=> x2 + y2 = 4p + 2, where p = n2 + m2 + n + m
=> Since 4p and 2 are even numbers, So 4p + 2 is an even number.
=> x2 + y2 is an even number and leaves the remainder when divided by 4
=> So, x2 + y2 is an even but not divisible by 4
6. HCF of two numbers is a factor of the LCM of thoses numbers.
Thus, we can not have two numbers whose HCF is 16 and LCM is 380
This is because when we divide 380 by 16, we get a remainder of 12
Thus 16 is not a factor of 380
So, we can not have two numbers whose HCF is 16 and LCM is 380
7. First, we find the LCM of 35, 56 and 91
35 = 5 * 7
56 = 2 * 2 * 2 * 7
91 = 7 * 13
LCM(35, 56, 91) = 2 * 2 * 2 * 5 * 7 * 13 = 3640
Now, add 7 to this number, we get 3647
So, the required number is 3647
NOTE: Please DO NOT ask multiple parts in one questions.
Please ask the remaining questions as separate questions.
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