Number System is one of the most important topics in the Quant Section of the CAT Exam. You can check out these Number System\u00a0 in<\/span> CAT Previous year papers<\/a>. <\/strong><\/span>In this Section, you can watch these Number System basics<\/strong><\/span>. This article will look into some important Number System Questions for CAT. These are good sources for practice; If you want to practice these questions, you can download this CAT Number System Questions PDF below, which is completely Free.<\/p>\n Download Number System Questions for CAT<\/a><\/p>\n Download Data Interpretation Questions for CAT<\/a><\/p>\n Question 1:\u00a0<\/b>What are the last two digits of $7^{2008}$?<\/p>\n a)\u00a021<\/p>\n b)\u00a061<\/p>\n c)\u00a001<\/p>\n d)\u00a041<\/p>\n e)\u00a081<\/p>\n 1)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n $7^4$ = 2401 = 2400+1 Question 2:\u00a0<\/b>A shop stores x kg of rice. The first customer buys half this amount plus half a kg of rice. The second customer buys half the remaining amount plus half a kg of rice. Then the third customer also buys half the remaining amount plus half a kg of rice. Thereafter, no rice is left in the shop. Which of the following best describes the value of x?<\/p>\n a)\u00a0$2 \\leq x \\leq 6$<\/p>\n b)\u00a0$5 \\leq x \\leq 8$<\/p>\n c)\u00a0$9 \\leq x \\leq 12$<\/p>\n d)\u00a0$11 \\leq x \\leq 14$<\/p>\n e)\u00a0$13 \\leq x \\leq 18$<\/p>\n 2)\u00a0Answer\u00a0(B)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n After the first sale, the remaining quantity would be (x\/2)-0.5 and after the second sale, the remaining quantity is 0.25x-0.75<\/p>\n After the last sale, the remaining quantity is 0.125x-(7\/8) which will be equal to 0<\/p>\n So\u00a00.125x-(7\/8) = 0 => x = 7<\/p>\n Question 3:\u00a0<\/b>How many even integers n, where $100 \\leq n \\leq 200$ , are divisible neither by seven nor by nine?<\/p>\n a)\u00a040<\/p>\n b)\u00a037<\/p>\n c)\u00a039<\/p>\n d)\u00a038<\/p>\n 3)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Between 100 and 200 both included there are 51 even nos. There are 7 even nos which are divisible by 7 and 6 nos which are divisible by 9 and 1 no divisible by both. hence in total 51 – (7+6-1) = 39<\/p>\n There is one more method through which we can find the answer. Since we have to find even numbers, consider the numbers which are divisible by 14, 18 and 126 between 100 and 200. These are 7, 6 and 1 respectively.<\/p>\n Question 4:\u00a0<\/b>The number of positive integers n in the range $12 \\leq n \\leq 40$ such that the product (n -1)*(n – 2)*\u2026*3*2*1 is not divisible by n is<\/p>\n a)\u00a05<\/p>\n b)\u00a07<\/p>\n c)\u00a013<\/p>\n d)\u00a014<\/p>\n 4)\u00a0Answer\u00a0(B)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n positive integers n in the range $12 \\leq n \\leq 40$ such that the product (n -1)*(n – 2)*\u2026*3*2*1 is not divisible by n, implies that n should be a prime no. So there are 7 prime nos. in given range. Hence option B.<\/p>\n Question 5:\u00a0<\/b>Let T be the set of integers {3,11,19,27,\u2026451,459,467} and S be a subset of T such that the sum of no two elements of S is 470. The maximum possible number of elements in S is<\/p>\n a)\u00a032<\/p>\n b)\u00a028<\/p>\n c)\u00a029<\/p>\n d)\u00a030<\/p>\n 5)\u00a0Answer\u00a0(D)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n No. of terms in series T , 3+(n-1)*8 = 467 i.e. n=59.<\/p>\n Now S will have atleast have of 59 terms i.e 29 .<\/p>\n Also the sum of 29th term and 30th term is less than 470.<\/p>\n Hence, maximum possible elements in S is 30.<\/p>\n Checkout: CAT Free Practice Questions and Videos<\/p>\n Question 6:\u00a0<\/b>The sum of four consecutive two-digit odd numbers, when divided by 10, becomes a perfect square. Which of the following can possibly be one of these four numbers?<\/p>\n a)\u00a021<\/p>\n b)\u00a025<\/p>\n c)\u00a041<\/p>\n d)\u00a067<\/p>\n e)\u00a073<\/p>\n 6)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Maximum sum of the four numbers <= 384=99+97+95+93 Question 7:\u00a0<\/b>The digits of a three-digit number A are written in the reverse order to form another three-digit number B. If B > A and B-A is perfectly divisible by 7, then which of the following is necessarily true?<\/p>\n a)\u00a0100<A<299<\/p>\n b)\u00a0106<A<305<\/p>\n c)\u00a0112<A<311<\/p>\n d)\u00a0118<A<317<\/p>\n 7)\u00a0Answer\u00a0(B)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Let A = 100x + 10y + z \u00a0and B = 100z + 10y + x .According to given condition\u00a0B – A = 99(z – x) As (B – A) is divisible by 7 . So clearly \u00a0(z – x) should be\u00a0\u00a0divisible by 7.\u00a0\u00a0z and x can have values 8,1 or 9,2 , such that 8-2=9-2=7 and\u00a0\u00a0y can have \u00a0value from 0 to 9. Question 8:\u00a0<\/b>For a positive integer n, let $P_n$ denote the product of the digits of n, and $S_n$ denote the sum of the digits of n. The number of integers between 10 and 1000 for which $P_n$ + $S_n$ = n is<\/p>\n a)\u00a081<\/p>\n b)\u00a016<\/p>\n c)\u00a018<\/p>\n d)\u00a09<\/p>\n 8)\u00a0Answer\u00a0(D)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Let n can be a 2 digit or a 3 digit number.<\/p>\n First let\u00a0n be a 2 digit number. Now if\u00a0n is\u00a0a 3 digit number. Hence option D.<\/p>\n Question 9:\u00a0<\/b>Let S be a set of positive integers such that every element n of S satisfies the conditions a)\u00a09<\/p>\n b)\u00a010<\/p>\n c)\u00a011<\/p>\n d)\u00a012<\/p>\n 9)\u00a0Answer\u00a0(A)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n The no. has all the digits as odd no. and is divisible by 3. So the possibilities are<\/p>\n 1113 Question 10:\u00a0<\/b>Of 128 boxes of oranges, each box contains at least 120 and at most 144 oranges. X is the maximum number of boxes containing the same number of oranges. What is the minimum value of X?<\/p>\n a)\u00a05<\/p>\n b)\u00a0103<\/p>\n c)\u00a06<\/p>\n d)\u00a0Cannot be determined<\/p>\n 10)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Each box contains at least 120 and at most 144 oranges.<\/p>\n So boxes may contain 25 different numbers of oranges among 120, 121, 122, …. 144.<\/p>\n Lets start counting.<\/p>\n 1st 25 boxes contain different numbers of oranges and this is repeated till 5 sets as 25*5=125.<\/p>\n Now we have accounted for 125 boxes. Still 3 boxes are remaining. These 3 boxes can have any number of oranges from 120 to 144.<\/p>\n Already every number is in 5 boxes. Even if these 3 boxes have different number of oranges, some number of oranges will be in 6 boxes.<\/p>\n Hence the number of boxes containing the same number of oranges is at least 6.<\/p>\n Question 11:\u00a0<\/b>Let n be the number of different five-digit numbers, divisible by 4 with the digits 1, 2, 3, 4, 5 and 6, no digit being repeated in the numbers. What is the value of n?<\/p>\n a)\u00a0144<\/p>\n b)\u00a0168<\/p>\n c)\u00a0192<\/p>\n d)\u00a0None of these<\/p>\n 11)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n To be divisible by 4 , last 2 digits of the 5 digit no. should be divisible by 4 . So possibilities are 12,16,32,64,24,36,52,56 which are 8 in number. Remaining 3 digits out of 4 can be selected in $^4C_3 $ ways and further can be arranged in 3! ways . So in total = 8*4*6 = 192<\/p>\n Question 12:\u00a0<\/b>Let D be recurring decimal of the form, $D = 0.a_1a_2a_1a_2a_1a_2…$, where digits $a_1$ and $a_2$ lie between 0 and 9. Further, at most one of them is zero. Then which of the following numbers necessarily produces an integer, when multiplied by D?<\/p>\n a)\u00a018<\/p>\n b)\u00a0108<\/p>\n c)\u00a0198<\/p>\n d)\u00a0288<\/p>\n 12)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Case 1: $a_1=0$ Case 2: $a_2=0$ So, in both the cases, 99D is an integer. From the given options, only option C satisfies this condition (198=2*99) and hence the correct answer is C.<\/p>\n Question 13:\u00a0<\/b>If $x^2 + y^2 = 0.1$ and |x-y|=0.2, then |x|+|y| is equal to:<\/p>\n a)\u00a00.3<\/p>\n b)\u00a00.4<\/p>\n c)\u00a00.2<\/p>\n d)\u00a00.6<\/p>\n 13)\u00a0Answer\u00a0(B)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n $(x – y)^2 = x^2 + y^2 – 2xy$<\/p>\n $0.04 = 0.1 – 2xy => xy = 0.03$<\/p>\n So, |xy| = 0.03<\/p>\n $(|x| + |y|)^2 = x^2 + y^2 + 2|xy| = 0.1 + 0.06 = 0.16$<\/p>\n So, |x|+|y| = 0.4<\/p>\n Question 14:\u00a0<\/b>What is the greatest power of 5 which can divide 80! exactly?<\/p>\n a)\u00a016<\/p>\n b)\u00a020<\/p>\n c)\u00a019<\/p>\n d)\u00a0None of these<\/p>\n 14)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n The highest power of 5 in 80! = [80\/5] + [$80\/5^2$] = 16 + 3 = 19<\/p>\n So, the highest power of 5 which divides 80! exactly = 19<\/p>\n Question 15:\u00a0<\/b>If x is a positive integer such that 2x +12 is perfectly divisible by x, then the number of possible values of x is<\/p>\n a)\u00a02<\/p>\n b)\u00a05<\/p>\n c)\u00a06<\/p>\n d)\u00a012<\/p>\n 15)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n If 2x+12 is perfectly divisible by x, then 12 must be divisible by x. Question 16:\u00a0<\/b>If a number 774958A96B is to be divisible by 8 and 9, the respective values of A and B will be<\/p>\n a)\u00a07 and 8<\/p>\n b)\u00a08 and 0<\/p>\n c)\u00a05 and 8<\/p>\n d)\u00a0None of these<\/p>\n 16)\u00a0Answer\u00a0(B)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n According to the divisible rule of 9, the\u00a0sum of all digits should be divisible by 9. Question 17:\u00a0<\/b>If n is an integer, how many values of n will give an integral value of $\\frac{(16n^2+ 7n+6)}{n}$ ?<\/p>\n a)\u00a02<\/p>\n b)\u00a03<\/p>\n c)\u00a04<\/p>\n d)\u00a0None of these<\/p>\n 17)\u00a0Answer\u00a0(D)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n Expression can be reduced to 16n + 7 + $\\frac{6}{n}$ Question 18:\u00a0<\/b>$n^3$ is odd. Which of the following statement(s) is\/are true? a)\u00a0I only<\/p>\n b)\u00a0II only<\/p>\n c)\u00a0I and II<\/p>\n d)\u00a0I and III<\/p>\n 18)\u00a0Answer\u00a0(C)<\/strong><\/p>\n View Video Solution<\/a><\/p>\n Solution:<\/b><\/p>\n if $n^3$ is odd then $n$ will be odd. let’s say it is $2k+1$ Question 19:\u00a0<\/b>How many five digit numbers can be formed from 1, 2, 3, 4, 5, without repetition, when the digit at the unit\u2019s place must be greater than that in the ten\u2019s place?<\/p>\n a)\u00a054<\/p>\n b)\u00a060<\/p>\n c)\u00a017<\/p>\n d)\u00a02 \u00d7 4!<\/p>\n
\nSo, any multiple of $7^4$ will always end in 01
\nSince 2008 is a multiple of 4, $7^{2008}$ will also end in 01<\/p>\n
\n384\/10 = 38.4
\nSo, the perfect square is a number less than 38.4
\nThe possibilities are 36, 25, 16 and 9
\nFor the sum to be 360, the numbers can be 87, 89, 91 and 93
\nThe sum of four consecutive odd numbers cannot be 250
\nFor the sum to be 160, the numbers can be 37,39,41 and 43
\nThe sum of 4 consecutive odd numbers cannot be 90
\nSo, from the options, the answer is 41.<\/p>\n
\nSo Lowest possible value of A lowest x,y and z which is \u00a0is 108 and the highest possible value of A is 299.<\/p>\n
\nSo n = 10x + y and Pn = xy and Sn = x + y
\nNow, Pn + Sn = n
\nTherefore, xy + x + y = 10x + y , we have y = 9 .
\nHence there are 9 numbers 19, 29,..\u00a0,99, so 9 cases .<\/p>\n
\nLet n = 100x + 10y + z
\nSo Pn = xyz and Sn = x + y + z
\nNow, for Pn + Sn = n ; \u00a0xyz + x + y + z = 100x + 10y + z ; so.\u00a0xyz = 99x + 9y .
\nFor above equation there is no value for which the above equation have an integer (single\u00a0digit) value.<\/p>\n
\nA. 1000 <= n <= 1200
\nB. every digit in n is odd
\nThen how many elements of S are divisible by 3?<\/p>\n
\n1119
\n1131
\n1137
\n1155
\n1173
\n1179
\n1191
\n1197
\nHence 9 possibilities .<\/p>\n
\nSo, D equals $0.0a_20a_20a_2…$
\nSo, 100D equals $a_2.0a_20a_2…$
\nSo, 99D equals $a_2$<\/p>\n
\nSo, D\u00a0equals $0.a_10a_10a_1…$
\nSo, 100D equals $a_10.a_10a_1….$
\nSo, 99D equals $a_10$<\/p>\n
\nHence, there are six possible values of x : (1,2,3,4,6,12)<\/p>\n
\ni.e. 55+A+B = 9k
\nSo sum can be either 63 or 72.
\nFor 63, A+B should be 8.
\nIn given options, option B has values of A and B whose sum is 8 and by putting them we are having a number which divisible by both 9 and 8.
\nHence answer will be B.<\/p>\n
\nNow to make above value \u00a0an integer n can be 1,2,3,6,-1,-2,-3,-6
\nHence answer will be D).<\/p>\n
\nI. $n$ is odd.
\nII.$n^2$ is odd.
\nIII.$n^2$ is even.<\/p>\n
\nthen $n^2$ will be = $(4k^2 + 4k + 1)$ which will be odd
\nHence answer will be C.<\/p>\n