Number System Test - 5

The data is inconsistent as the LCM is always a multiple of HCF. The product of two numbers is always equal to the product of their LCM and HCF. Here, HCF = 75 and LCM = 216 do not show consistency, i.e. no such number is possible.

72 * 12 = 24 * n ==> n = 36

By divisibility rule of 3, 4 + 5 + a + 6 = multiple of 3 ⇒ a =3 or 6 or 9 ⇒ by divisibility rule of 11⇒ (4 + 5) - (6 + a) = multiple of 11n or zero⇒ A = 3.

A number leaving 3 as remainder when divide by 5 ends either 3 or 8 .The number when squared will always have the last digit 4 or 9, thus leaving a remainder of 4 when divisible by 5.

The best way is to check the options. (1) is not possible
(3) 1/24 x 48 = 2, cube of 2 = 8 which is not a 2 digit number
(2) 1/24 x 72 = 3, cube of 3 = 27 which is reverse of 72.

3¹ = 3, 3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243, 3⁶ = 729, 3⁷ = 2167, 3⁸ = 6501.......

From the above expressions, we can observe that the unit digit of the result gets repeated after every four consecutive results. So, the unit digit of 3⁶⁸ will be 1.

Hence, the unit digit of 3⁶⁹ will be 3.
Similarly, the unit digit of 6⁸⁶⁴ will be unit digit of 2⁸⁶⁴ × 3⁸⁶⁴, which is 6 × 1 = 6.

And, unit digit of 4⁷²⁵ will be 4 (as 725 = 2n + 1).

Suppose that the last two digits of the four-digit number are c and d, where c is at the tens place and d is at the units place.
We can ignore the first two digits.

This gives us,
(10c + d) - (10d + c) = 54

9(c - d) = 54
c - d = 6

aaaaaa = aaa * 1000 + aaa
aaa (1000 + 1) = 1001(aaa)
=7 * 11 *13 (aaa)
This is obviously divisible by 7,11 and 13.

Only option 1 is divisible by three.

The best way is to factorise the number i.e 385 = 5 x 7 x 11
1001 = 7 x 11 x 13

So, the four prime numbers are 5, 7, 11 and 13.
Hence, first prime number = 5