Cambridge IGCSE Mathematics - International 0607 — 2014 May/June Paper 5 · Variant 1

This document consists of 6 printed pages and 2 blank pages. IB14 06_0607_51/3RP © UCLES 2014 [Turn over *6286435950* Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) May/June 2014 1 hour Candidates answer on the Question Paper. Additional Materials: Graphics Calculator READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. Do not use staples, paper clips, glue or correction fluid. You may use an HB pencil for any diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer all the questions. You must show all relevant working to gain full marks for correct methods, including sketches. In this paper you will also be assessed on your ability to provide full reasons and communicate your mathematics clearly and precisely. At the end of the examination, fasten all your work securely together. The total number of marks for this paper is 24.

2 © UCLES 2014 0607/51/M/J/14 Answer all the questions. INVESTIGATION COUNTING FACTORS This investigation looks for a method to find how many factors a number has. 1 The five factors of 16 are: 1, 2, 4, 8, 16 Write down the five factors of 16 as powers of 2. Three have been written down for you. 20 , 21 , , , 24 2 (a) Write down, in ascending order, the four factors of 27. Two have been written down for you. 1 , , , 27 (b) Write down the four factors of 27 as powers of 3. , , , 3 p is a prime number. (a) Write down the six factors of p5 as powers of p. Two have been written down for you. p0 , , , , , p5 (b) Write down, in terms of n, the number of factors of pn.

3 © UCLES 2014 0607/51/M/J/14 [Turn over 4 (a) 2 is a prime number and 128 = 27. Write down the number of factors of 128. (b) Work out each factor of 128 as an ordinary number. 5 5 is a prime number. (a) Write 125 as a power of 5. (b) Write down the number of factors of 125. 6 (a) A number, which can be written as a power of 2, has exactly 14 factors. Find this number. (b) Find another number that has exactly 14 factors.

4 © UCLES 2014 0607/51/M/J/14 7 (a) 20 is not a prime number. 20 = 22 × 51 where 2 and 5 are prime numbers. Find all the factors of 20 by completing the table. Powers of 5 50 51 Powers of 2 20 20 × 50 = 1 × 1 = 1 20 × 51 = …… × …… = …… 21 21 × 50 = 2 × 1 = 2 21 × 51 = …… × …… = …… 22 22 × 50 = …… × …… = …… 22 × 51 = 4 × 5 = 20 (b) The table has 3 rows and 2 columns. Describe how to find the number of factors of 20 from the number of rows and the number of columns. (c) 784 = 24 × 72 where 2 and 7 are prime numbers. (i) How many rows and how many columns would there be in the table for 784? rows columns (ii) Work out the number of factors of 784. Do not write them out.

5 © UCLES 2014 0607/51/M/J/14 [Turn over 8 (a) 1000 = 2n × 5n . Find n. (b) Use the method of question 7(c) to find the number of factors of 1000. (c) Use the method of part (a) and part (b) to find the number of factors of 1 000 000.

6 © UCLES 2014 0607/51/M/J/14 9 (a) 85 = 5 × 17 where 5 and 17 are prime numbers. Use the method of question 7(c) to show why 85 has exactly 4 factors. (b) Find all the numbers that are greater than 80 but smaller than 90 and have exactly 4 factors. This list of prime numbers is useful in answering this question. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89

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CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2014 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core), maximum raw mark 24 This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 51 © Cambridge International Examinations 2014 1 22, 23 1 2 (a) (b) 3, 9 30, 3[1], 32, 33 1 1 3 (a) (b) p[1], p2 , p3 , p4 n + 1 1 1 4 (a) (b) 8 1, 2, 4, 8, 16, 32, 64, 128 1 1 C opportunity 5 (a) (b) 53 4 1 1FT FT their power in (a) + 1. 6 (a) (b) 8 192 1 594 323 or 1 220 703 125 or other prime13 evaluated 1 1 C opportunity C opportunity 7 (a) (b) (c) (i) (ii) Multiply [3 by 2] oe 5 3 15 Powers of 5 50 51 Power of 2 20 20×50 = 1×1 = 1 20×51 = 1×5 = 5 21 21×50 = 2×1 = 2 21×51= 2×5 = 10 22 22×50 = 4×1 = 4 22×51= 4×5 = 20 2 1 1 1FT B1 for 1 correct cell. Do not accept with incorrect numbers FT their (c)(i)’s multiplied 8 (a) (b) (c) 3 soi 16 49 1 1FT 2 FT their n in (a) ≠ 0, 1 C opportunity M1 for 106 or 26×56 or 10002 seen. C opportunity

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 51 © Cambridge International Examinations 2014 9 (a) (b) 51, 171 and 2 × 2 soi 82, [85], 86, 87 1 2 B1 for one –1 for each extra between 80 and 90. SC1 3 correct and 2 wrong C opportunity Communication seen in one of the following questions 4(b), 6(a), 6(b), 8(b), 8(c), 9(b) 1