2026/070) Simplify $\sqrt{8+\sqrt{7}} - \sqrt{8-\sqrt{7}}$

Because $8 + 2 \sqrt(7)$ has $\sqrt(7)$ as one of the terms so the square root is of the form $a + b \sqrt(7)$ where a and b are rational numbers

Squaring we get

$a^2 + 7b^2 + 2ab \sqrt(7) = 8 + 2 \sqrt(7)$

So comparing rational and irrational parts we get

$a^2 + 7b^2 = 8\cdots(1)$

and $ab = 1\cdots(2)$

So a and b both are positive (-ve shall give -ve square root)

From (2) we get

$b = \frac{1}{a}$

Putting in (1) we get

$a^2 + 7 \frac{1}{a^2} = 8$

or $a^4 - 8a^2 +7=0$

or $(a^2-1)(a^2-7) = 0$

as a is rational so $a^2-7=0$ is ruled out and we have $a^2-1=0$

As a is positive a = 1 and so b = 1 from (2)

So $\sqrt{8+\sqrt{7}} = 1+ \sqrt{7}\cdots(3)$

Now Similarly $\sqrt{8 -\sqrt{7}} = \pm (1- \sqrt{7})$ we need to chooses the proper sign  

We need to take the principal root that is the value

 $\sqrt{8 -\sqrt{7}} = \sqrt{7}-1\cdots(4)$

from (3) and (4)

 $\sqrt{8+\sqrt{7}} - \sqrt{8-\sqrt{7}}= 2$

 

 

2026/069) Find the smallest positive three-digit integer n such that $3^n+4^n$ is divisible by 5.

We need to find $3^n+ 4^n \mod 5$

Let us take the $3^n \mod 5$ and see the behaviour As 5 is prime we have

$3^4 \equiv 1 \mod 5$

and also

$4^4 \equiv 1 \mod 5$

so $3^n + 4^n \mod 5 $ have period  4

Let us consider $f(n) = 3^n + 4^n \pmod 5$

We have

$f(0) = 2$

$f(1) = 2$

$f(2) = 0$

$f(3) = 1$

And $f(n) = f(4k+n)$

We see that $f(2)$ is zero so $f(4k+2)$ is zero

So we need to find the smallest k suck that $4k + 2 > 99$ and get $k = 25$ and smallest number $102$

 

2026/068) What is the smallest positive integer that has exactly 24 factors

Basis

To find the number of positive divisors of an integer, find its prime factorization, add one to each exponent, and multiply those sums together.

That is If $N=p_1^{q_1}p_2^{q_2}\cdots p_n^{q_n}$

it shall have $(q_1+1) (q_2+1)\cdots(q_n+1)$ factors

kindy note that $p_1,p_2\cdots p_n$ are relatively prime and not $q_1+1,q_2+1\cdots q_n+1$ 

Solution

Let is factor 24 in as many ways as we can and put N as product of power of primes and for the number to be lower higher power shall be with lower numer

 24 = 24 this gives number $2^23$

24 = 8 * 3 giving $2^7 *3^2= 1152 $=  this is smaller

can we make it smaller 

24 = 4 * 2 *3 = 4 * 3 * 2 giving $ 2^3 *3^2 * 5 = 360$ 

if we try to make a smaller number we get 3 * 2 * 2 * 2 giving $2^2 * 3 * 5 * 7 = 420$ and it is larger

so the ans is $360$   

 

2026/067) Given that $ \sum_{k=1}^{35}\sin\,5k =\tan(\frac{a}{b}) $ where angles are measured in degrees and a and b are relatively prime positive integers that satisfy $\frac{a}{b} < 90$ evaluate a + b

We shall use the formula for sum of sin of angles in AP

We have

$ \sum_{k=1}^{n}\sin\,kx = \dfrac{\sin\frac{nx}{2}\sin\frac{(n+1)x}{2}}{\sin \frac{x}{2}}$

Put n = 35 and $ x = 5^\circ$

To get 

 $ \sum_{k=1}^{35}\sin\,5k^\circ = \frac{\sin\dfrac{175}{2}^\circ\sin\,90^\circ}{\sin \dfrac{5}{2}^\circ}$

$ = \dfrac{\sin\dfrac{175}{2}^\circ}{\cos \dfrac{175}{2}^\circ}=\tan  \dfrac{175}{2}^\circ$

From given condition $a=175,b=2,\dfrac{a}{b} = 87.5 <90$ 

So $a+b = 177$

2026/066) Determine all triples (a,b,c) of positive integers with $a^{b^c} = (a^b)^c$

 We are given $a^{b^c} = a^{bc}$

Or $a^{b^c-bc} = 1$
 

This has got 2 sets of solutions
 

1) a = 1 and b and c can be any positive integer
 

2) a = any positive integer other than 1 and
 

$b^c - bc = 0$
 

Or $b(b^{c-1}) = c$
 

As b is not zero so we have $ b^{c-1} = c $
 

if c = 1 we get b any number
 

Or $b= \sqrt[c-1]c$ $c>=2$
 

The root can be taken to be integer if c-1 = 1 or c =2 giving b = 2
 

So solution set $(1,b,c)$ or $(a,2,2)$ or $(a,b,1)$ where a,b,c are any integers

2026/065) If x and y are positive real numbers that satisfy the equation $x+4\sqrt{xy} -2 \sqrt{x} - 4 \sqrt{y} + 4y =3$ evaluate $\frac{\sqrt{x} + 2\sqrt{y} + 2014}{ 4- \sqrt{x} - 2\sqrt{y}}$

Solution 

We are given

$x+4\sqrt{xy} -2 \sqrt{x} - 4 \sqrt{y} + 4y =3$ 

Adding 1 to both sides

$x+4\sqrt{xy} -2 \sqrt{x} - 4 \sqrt{y} + 4y + 1=4$

Or $ (\sqrt{x} +  2\sqrt{y} -1)^2 = 4$ or $ (\sqrt{x} +  2\sqrt{y} -1) = 2$ as both square roots are positive

So  $ (\sqrt{x} +  2\sqrt{y}) = 3$

So  $\frac{\sqrt{x} + 2\sqrt{y} + 2014}{ 4- \sqrt{x} -  2\sqrt{y}}=  \frac{3 + 2014}{ 4- 3} = 2017$
 

2026/064) Given a,b,c,d are roots of the equation $x^4−7x^3+3x^2−21x+1=0$ Evaluate $(a+b+c)(b+c+d)(c+d+a)(d+a+b)$

Because a,b ,c ,d are roots of the equation $x^4−7x^3+3x^2−21x+1=0$

We have by vieta's formula
 

$a+b+c+d = 7\cdots(1)$
 

Let
 

$f(x) = x^4−7x^3+3x^2−21x+1=(x-a)(x-b)(x-c)(x-d)\cdots(2)$
 

From (1) we have
 

$a+b+c = 7-d\cdots(3)$
 

$b+c+d  = 7 -a\cdots(4)$
 

$c+d+a = 7 -b\cdots(5)$
 

 $d+a +b = 7-c\cdots(6)$
 

From (3), (4), (5),(6) we have
 

$(a+b+c)(b+c+d)(c+d+a)(d+a+b)= (7-d)(7-a)(7- b)(7-c) = f(7)\cdots(7)$ from (2)
 

As $f(x) = x^4−7x^3+3x^2−21x+1$
 

So $f(7) = 7^4 - 7 * 7^3 +3 *7^2 - 21 * 7 + 1 = 1$
 

From (7) and above we have
 

$(a+b+c)(b+c+d)(c+d+a)(d+a+b) = 1$