Number System - Study Mode

[#61] If X and Y are the two digits of the number 347XY such that the number is completely divisible by 80, then what is the value of X + Y?
Correct Answer

(A) 2

Explanation

Solution: Given: 347XY where X, Y two digit number is completely divisible by 80. $$eqalign{
& frac{{347XY}}{{80}} = frac{{347XY}}{{8 imes 10}} cr
& frac{{7 + X + Y}}{8} cr} $$ ∴ Y must be zero So, go through option A Put X = 2, then number 3472 is divisible by 8.

[#62] When 12, 16, 18, 20 and 25 divide the least number x, the remainder in each case is 4 but x is divisible by 7. What is the digit at the thousand's place in x?
Correct Answer

(D) 8

Explanation

Solution: LCM of 12, 16, 18, 20, 25 = 3600 For remainder 4 = 3600k + 4 If divisible by 7 $$ = frac{{7 imes 514k + 2k + 4}}{7}$$ $$frac{{2k + 4}}{7},$$ xa0If k = 5 divisible x = 3600 × 5 + 4 = 18000 + 4 $$ = mathop {mathop {18004}limits_ downarrow }limits_{x08oxed8} $$

[#63] The value of (1018) 2 - 1019 × 1017 + 1015 × 1012 - 1016 × 1011 is:
Correct Answer

(D) 5

Explanation

Solution: 1018 2 - 1019 × 1017 + 1015 × 1012 - 1016 × 1011 = 1018 2 - (1018 + 1) × (1018 - 1) + (1016 - 1) × (1011 + 1) - 1016 × 1011 = 1018 2 - 1018 2 + 1 + 1016 × 1011 + 1016 - 1011 - 1 - 1016 × 1011 = 5

[#64] If the sum of digits of a two digit number is 10 and if the digits of two digits number is reversed, then the number is decreased by 36. Which of the following is correct regarding the number? I. The difference of the digits is 4. II. The value of number can be 84. III. Number is always a composite number.
Correct Answer

(D) I and II

[#65] When (77 77 + 77) is divided by 78, the remainder is:
Correct Answer

(C) 76

Explanation

Solution: $$eqalign{
& frac{{{{77}^{77}} + 77}}{{78}} cr
& = {left( { - 1}
ight)^{77}} + left( { - 1}
ight) cr
& = - 1 - 1 cr
& = - 2 cr
& { ext{Remainder}} = 78 - 2 = 76 cr} $$