A certain number of two digits is three times the sum of its digits and if 45 be added to it, then digits are reversed. The number is?

Let second number be 3N, then, first one is 6N and the third one is 2N.
From question 3N + 6N + 2N = 132
? 11N = 132
? N = 12

? Second number = 3N = 3 x 12 = 36

Let the value for ? be N
So N x 9 x 7 x 3 = a number with unit digit 1.
Clearly, the minimum value of N is 9.

Let x/5 = p and let y when divided by 5 gives q as quotient and 1 as remainder.
Then, y = 5q + 1
Now x = 5p and y = 5q + 1
? x + y = 5p + 5q + 1 = 5(p + q) + 1
So, required remainder = 1

Let the second number be N.
Then, first number = 2N and third number = 4N
? 2N + N+ 4N/3 = 56
? 7N = 3 x 56
? N = 3 x 56/7 = 24
So, the smallest number is 24.

A number that ends in 2 , 3 , 7 or 8 is never a perfect square.

Let the required fraction be p/q.
Then, (p +1) / (q +1) = 4
? p - 4q= 3 ....(i)

And, (p - 1) / (q -1) = 7
? p - 7q = -6 ....(ii)

Solving these equation we get p = 15, q=3

Let the number be a and b .
Then a² + b² = 68 ....(i)
and (a - b)² = 36
Now, (a - b)² = 36 ....(ii)
? a² + b² - 2ab = 36
? 68 - 2ab = 36
? 2ab = 32
? ab = 16 .

Let the numbers be x and y ,
Then, 2x + 3y = 18 ....(i),
3x + 2y = 17 ....(ii)

Solving, we get x =3 , y= 4
? Largest number = 4

Let the number be x, y, z
Then,
x + y = 45,
y + z = 55,
z + 3x = 30

Now, y = (45 - x) and
z = 55 - y
? z = 55 - (45 - x ) = 10 + x

? 10 + x + 3x = 90
? x = 20
So, third number = z
= 10 + x = 30

The number is of the form (5x + 3 ) where x is an integer

? [(5x + 3)²] / 5 = ( 25x² + 30x + 9) / 5
= 25x²/ 5 + 30x/5 + (5 +4)/5
? The remainder is 4.