{"id":12229,"date":"2025-02-13T15:31:32","date_gmt":"2025-02-13T10:01:32","guid":{"rendered":"https:\/\/cracku.in\/blog\/?p=12229"},"modified":"2025-02-21T11:31:48","modified_gmt":"2025-02-21T06:01:48","slug":"cat-questions-on-factors-of-a-number","status":"publish","type":"post","link":"https:\/\/cracku.in\/blog\/cat-questions-on-factors-of-a-number\/","title":{"rendered":"CAT Questions on Factors of a Number"},"content":{"rendered":"

CAT Questions on Factors of a Number:<\/strong><\/span><\/p>\n

Download CAT Questions on Factors of a Number<\/a><\/p>\n

CAT Online Coaching<\/a><\/p>\n

Download CAT Quant Questions PDF<\/a><\/p>\n

Take Free Mock Test for CAT 2023<\/a><\/p>\n

Question 1:<\/strong>\u00a0A \u201ctragic number\u201d is a number which can be expressed as the sum of three of its factors. For example, 6 can be expressed as the sum of 1, 2 and 3. How many tragic numbers are there that are less than 50?<\/p>\n

a) 6
\nb) 7
\nc) 8
\nd) 9<\/p>\n

Question 2:\u00a0<\/strong>If N = 1980, Find the number and sum of its even factors.<\/p>\n

a) 28, 6552
\nb) 24, 5616
\nc) 24, 4630
\nd) 28, 5672<\/p>\n

Question 3:<\/strong>\u00a0How many integers are both multiples of $125^{3124}$ and factors of $125^{3127}$?<\/p>\n

a) 3
\nb) 10
\nc) 4
\nd) 11<\/p>\n

Download CAT 2023 Syllabus PDF<\/a><\/p>\n

Question 4:<\/strong>\u00a0What is the number of even factors of 36000 which are divisible by 9 but not by 36?<\/p>\n

a) 20
\nb) 4
\nc) 10
\nd) 12<\/p>\n

Question 5:\u00a0<\/strong>How many factors of 36288 are perfect cubes?<\/p>\n

a) 9
\nb) 4
\nc) 6
\nd) 8<\/p>\n

Factors of a number – Formulas for CAT<\/a><\/p>\n

Formulas on Number system and factorials\u00a0<\/a><\/p>\n

Solutions for\u00a0CAT Questions on Factors of a Number:<\/strong><\/span><\/p>\n

Solutions:<\/strong><\/p>\n

1) Answer (C)<\/strong><\/p>\n

The factors of a number \u2018x\u2019 can be of the form $\\frac{x}{2}$ , $\\frac{x}{3}$ , $\\frac{x}{5}$ , \u2026.
\nIf 2 is not a factor of the number then the highest three factors of the number can be $\\frac{x}{3}$, $\\frac{x}{5}$, $\\frac{x}{7}$. The sum of these three is less than x. So 2 has to be a factor of the number.
\nIf 3 is not a factor of the number the the highest three factors of the number can be $\\frac{x}{2}$, $\\frac{x}{4}$, $\\frac{x}{5}$. The sum of these three is less than x. So 3 has to be a factor of the number.
\nIf 2 and 3 are factors, 6 is also a factor.
\nAlso the sum of $\\frac{x}{2}$, $\\frac{x}{3}$, $\\frac{x}{6}$ is exactly equal to x.
\nSo all numbers which are multiples of 6 are tragic numbers.
\nThere are 8 such numbers which are below 50.<\/p>\n

2) Answer (B)<\/strong><\/p>\n

1980 = $2^2 * 3^2 * 11 * 5$
\nNumber of even factors = Total number of factors – Number of odd factors.
\n= (2+1)(2+1)(1+1)(1+1) – (2+1)(1+1)(1+1) = 24
\nSum of even factors = Sum of all the factors – sum of odd factors
\n= $(\\frac {2^{(2+1)} -1} {(2-1)})$ x $(\\frac {3^{(2+1)} -1} {(3-1)})$ x $(\\frac {11^{(1+1)} -1} {(11-1)})$ x $(\\frac {5^{(1+1)} -1} {(5-1)})$ – $(\\frac {3^{(2+1)} -1} {(3-1)})$ x $(\\frac {11^{(1+1)} -1} {(11-1)})$ x $(\\frac {5^{(1+1)} -1} {(5-1)})$
\n= 7*13*12*6 – 13*12*6 = 5616<\/p>\n

Preparation tips for CAT – Beginners<\/a><\/p>\n

3) Answer (B)<\/strong><\/p>\n

$125^{3124}$ = $5^{9372}$
\n$125^{3127}$ = $5^{9381}$
\nThus, the factors of $5^{9381}$ which are multiples of $5^{9372}$ are $5^{9372}$,$5^{9373}$ … $5^{9381}$. Thus, there are 10 such integers<\/p>\n

4) Answer (B)<\/strong><\/p>\n

$36000 = 2^{5}*3^{2}*5^{3}$
\nSince we are talking of even factors, there must be at least one 2 in the required factors.
\nSince the number is divisible by 9, we must have both the threes.
\nWe cannot have more than 1 two as it will make the number divisible by 36.
\nSo we have 1 way of choosing 2, 1 way of choosing 3, 4 ways of choosing 5.
\nThus the required number of factors are
\n1*1*4 = 4<\/p>\n

5) Answer (C)<\/strong><\/p>\n

$36288 = 2^6 * 3^4 * 7$
\nFor any perfect cube, all the powers of its prime numbers have to be multiples of 3.
\nSo, if the factor is of form $2^a * 3^b * 7^c$, a can take values 0, 3, 6
\nAnd b can take values 0, 3
\nAnd c can take value 0.
\n==> There are 3*2*1 = 6 possibilities.<\/p>\n

CAT Mock Tests<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

CAT Questions on Factors of a Number: Download CAT Quant Questions PDF Take Free Mock Test for CAT 2023 Question 1:\u00a0A \u201ctragic number\u201d is a number which can be expressed as the sum of three of its factors. For example, 6 can be expressed as the sum of 1, 2 and 3. How many tragic […]<\/p>\n","protected":false},"author":21,"featured_media":12233,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_mi_skip_tracking":false,"footnotes":""},"categories":[3,169,125],"tags":[6106,144],"class_list":{"0":"post-12229","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-cat","8":"category-downloads","9":"category-featured","10":"tag-cat-2023","11":"tag-cat-exam"},"better_featured_image":{"id":12233,"alt_text":"CAT Questions on Factors of a Number","caption":"CAT Questions on Factors of a Number","description":"CAT Questions on Factors of a 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