Factors Notes (Form 1 Mathematics) | YNetStudyHub
Mathematics Hub Form 1 • KCSE Revision Notes
Maswali ya KCSE
Inasoma makala kwa sauti...
Karatasi ya Mtihani
KCSE 102/2 (Lugha)
Swali Maalumu
Swali la 4 (Sehemu D)
Ugawaji wa Alama
Alama 10 Kamili (12.5%)
Muda Unaopendekezwa
Dakika 20 – 25

Factors

Mtaala Sanifu wa KCSE Form 1 • Mathematics 3 min kusoma (Maneno 506) 49 waliosoma Imesasishwa Jul 2026
Yaliyomo Kwenye Makala Hii (Quick Jump):
Kidokezo cha Mtahini wa KCSE (KNEC Examiner Note)

Katika mtihani wa KCSE, kamwe usitaje sifa ya kiisimu bila kuthibitisha kwa dondoo au mfano kutoka kwenye kifungu cha swali! Mfano: Kuandika tu "Lugha ya heshima" hupewa alama sifuri. Lazima uandike: "Lugha ya heshima; msemaji anasema 'Mheshimiwa Hakimu' kuonyesha unyenyekevu." (Alama kamili).

Introduction

In mathematics, factors are whole numbers that can be multiplied together to give another number. Understanding factors is crucial in various mathematical operations, including simplifying fractions, finding common factors, and factoring polynomials. In this topic, we will explore the concept of factors in detail and learn how to identify, calculate, and work with factors.

Prime Numbers

Definition: A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself.

Example: Let's consider the number 7. The factors of 7 are 1 and 7 because these are the only two numbers that can multiply to give 7.

Worked Example: Find the factors of 17. $17$ is a prime number, so its factors are $1$ and $17$.

Composite Numbers

Definition: A composite number is a natural number greater than 1 that can be formed by multiplying two smaller natural numbers.

Example: 6 is a composite number because it can be expressed as $2 \times 3$.

Worked Example: Find the factors of 12. $12 = 1 \times 12$ and $12 = 2 \times 6$. Therefore, the factors of 12 are 1, 2, 3, 4, 6, and 12.

Prime Factorization

Definition: Prime factorization is the process of expressing a number as a product of its prime factors.

Example: The prime factorization of 24 is $2^3 \times 3$.

Worked Example: Find the prime factorization of 36. $36 = 2^2 \times 3^2$. Therefore, the prime factorization of 36 is $2^2 \times 3^2$.

Common Factors and Greatest Common Factor (GCF)

Definition: Common factors of two or more numbers are the numbers that divide each of the numbers evenly. The greatest common factor (GCF) is the largest of these common factors.

Example: The common factors of 12 and 18 are 1, 2, 3, and 6. The GCF of 12 and 18 is 6.

Worked Example: Find the GCF of 24 and 36. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24. The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36. The common factors are 1, 2, 3, 4, 6, and 12. Therefore, the GCF of 24 and 36 is 12.

Multiples

Definition: Multiples of a number are the result of multiplying that number by another integer.

Example: The multiples of 5 are 5, 10, 15, 20, etc.

Worked Example: Find the first five multiples of 4. The first five multiples of 4 are 4, 8, 12, 16, and 20.

Common Mistakes

  • Confusing prime numbers with composite numbers.
  • Forgetting to include 1 and the number itself as factors.
  • Incorrectly identifying the GCF by missing common factors.

Key Points

  • Prime numbers have only two factors: 1 and the number itself.
  • Composite numbers have more than two factors.
  • Prime factorization breaks down a number into its prime factors.
  • The GCF is the largest common factor of two or more numbers.
  • Multiples are numbers obtained by multiplying a number by integers.

Practice Questions

  1. Find the factors of 20.

Answer: The factors of 20 are 1, 2, 4, 5, 10, and 20.

  1. Determine the prime factorization of 48.

Answer: $48 = 2^4 \times 3$.

  1. What is the GCF of 36 and 48?

Answer: The GCF of 36 and 48 is 12.

  1. List the first four multiples of 7.

Answer: The first four multiples of 7 are 7, 14, 21, and 28.

  1. Find the GCF of 18 and 27.

Answer: The GCF of 18 and 27 is 9.

  1. Calculate the prime factorization of 75.

Answer: $75 = 3 \times 5^2$.

  1. Determine the factors of 42.

Answer: The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

  1. What are the common factors of 15 and 25?

Answer: The common factors of 15 and 25 are 1 and 5.

Practice these questions to strengthen your understanding of factors and their applications in various mathematical problems.

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Maswali ya Majaribio na Miongozo ya Majibu (Marking Schemes)

Jaribu kujibu maswali haya kwanza kisha ubofye "Onyesha Mwongozo wa Majibu" ili kuangalia ugawaji wa alama wa KNEC.

Swali la 1: Dondoo la Mahakamani (Alama 10 Kamili)

"Hati ya mashtaka inaeleza wazi kwamba mshtakiwa alipatikana na hatia ya kuvunja duka mnamo tarehe 12 mwezi uliopita. Shahidi nambari mbili alithibitisha kuwepo kwake eneo la tukio bila shaka yoyote. Upande wa utetezi unapinga ushahidi huo kwani mteja wangu alikuwa safarini siku hiyo..."

(a) Tambua sajili ya dondoo hili. (Alama 2)

(b) Fafanua sifa nne za kiisimu zinazojitokeza katika dondoo hili ukizingatia mifano kutoka kwenye kifungu. (Alama 8)

Mwongozo wa Kusahihisha wa KNEC:

(a) Sajili: Sajili ya Mahakamani / Mazingira ya Sheria. (Kutaja = Alama 2)

(b) Sifa 4 za kiisimu (Kila hoja alama 2: kutaja alama 1, kielelezo alama 1):

  1. Msamiati wa kisheria: Maneno kama hati ya mashtaka, mshtakiwa, hatia, shahidi, ushahidi, upande wa utetezi.
  2. Lugha ya heshima na taratibu: Utaratibu maalum wa uwasilishaji wa kesi.
  3. Lugha ya kupinga na kubishana: Upande wa utetezi unapinga ushahidi uliotolewa.
  4. Lugha kavu isiyo na utani: Mazungumzo yanazingatia uzito wa sheria bila hisia binafsi.
Swali la 2: Nadharia ya Lugha ya Taifa (Alama 6)

Eleza sababu sita zinazofanya lugha ya Kiswahili kutumiwa na wananchi wengi nchini Kenya kuliko Kiingereza. (Alama 6)

Mwongozo wa Majibu (Alama 1 kwa kila hoja sahihi):
  • Ni lugha ya taifa inayowaunganisha Wakenya wote bila ubaguzi wa kikabila.
  • Hutumika sana masokoni na katika biashara za kiholela mijini na vijijini.
  • Ina asili ya kibantu hivyo ni rahisi kujifunza kwa jamii nyingi za Kenya.
  • Vyombo vya habari (redio na runinga) vinatangaza kwa Kiswahili na kuenea kote.
  • Hufundishwa kama somo la lazima katika shule za msingi na upili.
  • Hutumika kwenye mikutano ya kisiasa, ibada makanisani/misikitini na michezo.
Swali la 3: Kuchanganya na Kubadili Msimbo (Alama 8)

(a) Tofautisha kati ya kuchanganya msimbo na kubadili msimbo. (Alama 2)

(b) Eleza sababu nne zinazowafanya watu wachanganye au wabadili msimbo katika mawasiliano. (Alama 6)

Mwongozo wa Majibu:

(a) Tofauti: Kuchanganya msimbo ni kutumia maneno ya lugha nyingine ndani ya sentensi ile ile, ilhali kubadili msimbo ni kubadili lugha nzima sentensi moja inapokamilika na kuanza nyingine kwa lugha tofauti. (Alama 2)

(b) Sababu 4:

  • Upungufu wa msamiati mwafaka katika lugha moja (hasa nyanja za kisasa za kiteknolojia au utabibu).
  • Kutaka kujitambulisha na kundi au tabaka fulani (k.m. tabaka la wasomi).
  • Kuficha siri ili watu wengine wasioelewa lugha hiyo wasipate ujumbe.
  • Kujenga ukaribu na urafiki au kuondoa ukiritimba katika maongezi.
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Mada Nyingine za Form 1 Mathematics

Endelea na masomo mengine ya mtaala wa Kidato cha Tatu:

Natural Numbers Integers Fractions Decimals Squares and Square Roots Algebraic Expressions Rates, Ratio, Proportion and Percentage Length Area Volume and Capacity Mass, Weight and Density Time
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