Number System - Study Mode
[#231] How many number are there between 1 to 200 which are divisible by 3 but not by 7?
Correct Answer
(C) 57
Explanation
Solution: Number from 1 to 200 which is divisible by 3 3, 6, . . . . . . . . ., 198 $$eqalign{
& = frac{{198 - 3}}{3} + 1 cr
& = 66 cr} $$ Total number which divisible by 3 & 7 $$eqalign{
& = frac{{189 - 21}}{{21}} + 1 cr
& = frac{{168}}{{21}} + 1 cr
& = 9 cr} $$ Total number which is divisible by 3 but not 7 = 66 - 9 = 57 Alternate solution Total number which is divisible by 3 & 7 both from 1 to 200 LCM (3 & 7) $$ o frac{1}{{21}} - frac{{200}}{{21}} = 0 - 9 = 9$$ Total number which is divisible by only 3, from 1 to 200 $$frac{1}{3} - frac{{200}}{3} = 0 - 66 = 66$$ ∴ Total number which is divisible by 3 but not 7 = 66 - 9 = 57
[#232] In a class of students, the first student has 2 toffees, second has 4 toffees. third has 6 toffees and so on. If the number of students in the class is 25. Then the total number of toffees are divisible by . . . . . .
Correct Answer
(B) 5 and 13
Explanation
Solution: $$eqalign{
& 2 + 4 + 6 + ...... cr
& {S_n} = frac{n}{2}left[ {2a + left( {n - 1}
ight)d}
ight] cr
& = frac{{25}}{2}left[ {2 imes 2 + 24 imes 2}
ight] cr
& = frac{{25}}{2} imes 52 cr
& = 25 imes 26 cr
& { ext{Clearly it is divisible by }}5,,13 cr} $$
[#233] Which of the following statement(s) is/are TRUE? I. The total number of positive factors of 72 is 12. II. The sum of first 20 odd numbers is 400. III. Largest two digit prime number is 97.
Correct Answer
(D) All are true
Explanation
Solution: I. N = 72 = 2 3 × 3 2 Total factors = (3 + 1)(2 + 1) = 4 × 3 = 12 True II. Sum of n odd number = n 2 Sum of 20 odd number = 20 2 = 400 True III. Largest 2 digit prime number is 97 True All are true
[#234] Nathu and Buchku each have certain number of oranges. Nathu says to Buckhu, "If you give me 10 of your oranges, I will have twice the number of oranges left with you". Buckhu replies, "If you give me 10 of your oranges, I will have the same number of oranges as left with you." What is the number of oranges with Nathu and Buckhu, respectively?
Correct Answer
(B) 70, 50
Explanation
Solution: Let the number of oranges with Nathu be x Number of oranges with Buchku = y Case I, x + 10 = 2(y - 10) ⇒ x + 10 = 2y - 20 ⇒ 2y - x = 20 + 10 = 30 . . . . . . (i) Case II, y + 10 = x - 10 ⇒ x - y = 10 + 10 = 20 . . . . . . (ii) On adding equations (i) and (ii) 2y - x + x - y = 30 + 20 ⇒ y = 50 From equation (ii), x - 50 = 20 ⇒ x = 50 + 20 = 70
[#235] The owner of a local jewellery store hired 3 watchmen to guard his diamonds, but a thief still got in and stole some diamonds. On the way out, the thief met each watchman, one at a time. To each he gave $$frac{1}{2}$$ of the diamonds he had then and 2 more besides. He escaped with one diamond. How many did he steal originally?
Correct Answer
(A) 36
Explanation
Solution: At last thief is left with one diamond. Hence, the number of diamonds before he gave some diamonds to the third watchman, $$eqalign{
& Rightarrow x - left( { {frac{x}{2}} + 2}
ight) = 1 cr
& { ext{or, }}{kern 1pt} frac{{ {x - 4} }}{2} = 1 cr
& { ext{or, }}{kern 1pt} x = 6 cr} $$ Hence, he had 6 diamonds before he gave 5 to the third watchman. Similarly number of diamonds before giving to second watchman, $$eqalign{
& frac{{ {x - 4} }}{2} = 6 cr
& { ext{or,}},{kern 1pt} x = 16 cr} $$ And number of diamonds before giving to the first watchman, $$eqalign{
& frac{{ {x - 4} }}{2} = 16 cr
& { ext{or, }}{kern 1pt} x = 36 cr} $$ ∴ The thief has stolen 36 diamonds originally