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L17 Modulo I

The document discusses the concept of congruence in integers, denoted by the expression a ≡ b (mod n), where a and b have the same remainder when divided by a non-zero integer n. It provides examples and practice problems related to finding values of integers based on congruence and divisibility. Additionally, it includes exercises involving remainders and sums of integers under certain conditions.

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27 views8 pages

L17 Modulo I

The document discusses the concept of congruence in integers, denoted by the expression a ≡ b (mod n), where a and b have the same remainder when divided by a non-zero integer n. It provides examples and practice problems related to finding values of integers based on congruence and divisibility. Additionally, it includes exercises involving remainders and sums of integers under certain conditions.

Uploaded by

chhs002
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Lesson 17 – Modulo I

Two integers 𝑎 and 𝑏 are said to be congruent if they are distinct numbers and have the
same remainder when they are divided by a non-zero integer, 𝑛. This is denoted by the
expression 𝑎 ≡ 𝑏 (𝑚𝑜𝑑 𝑛).

For example, since the distinct numbers 10 and 14 both give a remainder of 2 when
divided by 4, we say:

10 ≡ 14 (𝑚𝑜𝑑 4)

This also leads us to conclude the following,

𝑎 = 𝑘. 𝑛 + 𝑏

10 = 𝑘. 4 + 14, where 𝑘 = −1

and

𝑛|(𝑎 − 𝑏)

4|(10 − 14)

Example 1

Find a possible value of 𝑥 in the following expressions.

a) 2 ≡ 𝑥 (𝑚𝑜𝑑 9)

b) 11 ≡ 15 (𝑚𝑜𝑑 𝑥)

c) 23 ≡ 42 (𝑚𝑜𝑑 𝑥)

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Example 2

Given that (297 + 𝑘) is divisible by 31, find the value of 𝑘.

Practice

1. Given that (233 + 𝑚) is divisible by 31, find the value of 𝑚.

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2. Given that (269 + 𝑛) is divisible by 127, find the value of 𝑛.

Example 3

When the three numbers 958, 576 and 1340 are divided by a positive integer 𝑚, the
remainders are all equal to a positive integer 𝑛. Given that 𝑚 is a prime number, find the
value of (𝑚 + 𝑛).

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Practice

3. When the whole numbers 198, 295 and 877 are divided by a positive integer 𝑚, the
remainders are all equal to a positive integer 𝑛. Find (𝑚 + 𝑛).

4. Show that 111111 + 112112 + 113113 is divisible by 10.

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5. Find the remainder when 71427 × 19 is divided by 7.

Example 4

Find the remainder when (13113 + 13214 + 13315) is divided by 13.

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Practice

6. Today is a Saturday. On which day of the week is 365364 days later?

7. Find the remainder when 39795678 is divided by 39.

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8. Find the sum of all positive integers which are less than or equal to 200 and not
divisible by 3 or 5.

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9. Find the last four digits of 𝑚, where 𝑚 is the sum of the following series of numbers.

4, 44, 444, … , ⏟
444 … 4
2000

10. When an integral number 𝑚 is multiplied by 13, the last three digits is 123. Find the
smallest possible value of 𝑚.

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