Number System - Study Mode

[#516] The sum of 10 terms of the arithmetic series is 390. If the third term of the series is 19. Find the first term?
Correct Answer

(A) 3

Explanation

Solution: $$eqalign{
& { ext{Sum of A}}{ ext{.P}}{ ext{.}} = frac{n}{2}left[ {2a + left( {n - 1}
ight)d}
ight] cr
& {n^{{ ext{th}}}}{ ext{term}} = a + left( {n - 1}
ight)d cr
& {3^{{ ext{rd}}}}{ ext{term}} = a + left( {3 - 1}
ight)d = 19 cr
& a + 2d = 19,......,left( { ext{i}}
ight) cr
& { ext{Sum of 10 term}} = frac{{10}}{2}left[ {2a + 9d}
ight] = 390 cr
& 2a + 9d = 78,......,left( {{ ext{ii}}}
ight) cr
& a + 2d = 19,......,left( { ext{i}}
ight) cr
& { ext{From equation }}left( { ext{i}}
ight){ ext{ and }}left( {{ ext{ii}}}
ight) cr
& x08oxed{a = 3},,x08oxed{d = 8} cr} $$

[#517] If x = (164) 169 + (333) 337 - (727) 726 , then what is the units digit of x?
Correct Answer

(C) 8

Explanation

Solution: x = (164) 169 + (333) 337 - (727) 726 x = 4 + 3 - 9 x = . . . . . . 7 - 9 x = 17 - 9 x = 8

[#518] The quotient when 10 100 is divided by 5 75 is:
Correct Answer

(C) 2 75 × 10 25

Explanation

Solution: $$eqalign{
& frac{{{{10}^{100}}}}{{{5^{75}}}} cr
& = frac{{{2^{100}} imes {5^{100}}}}{{{5^{75}}}} cr
& = {2^{100}} imes {5^{25}} cr
& = {2^{25}} imes {2^{75}} imes {5^{25}} cr
& = {2^{75}} imes {10^{25}} cr} $$

[#519] x, y and z are distinct prime numbers where x < y < z. If x + y + z = 70, then what is the value of z?
Correct Answer

(C) 37

Explanation

Solution: Since x, y and z are distinct prime number (x < y < z) y + z = 70 - 2 = 68 31 + 37 = 68 Hence, x = 2, y = 31 and $$x08oxed{{ ext{z}} = 37}{ ext{ Answer}}$$

[#520] The number of factors of 196 which are divisible by 4 is:
Correct Answer

(D) 3

Explanation

Solution: 196 = 7 2 × 2 2 Factors ⇒ (1, 7, 49) × (1, 2, 4) Divisible by 4 = 4, 28, 196 = 3 factors