5. Working modulo 9 or 11, find the missing digits in the calculations below: (a) 51840 273581 = 1418243x040. (b) 2x99561 = [3(523 + x)]². (c) 2784x = x 5569. %3D %3D
5. Working modulo 9 or 11, find the missing digits in the calculations below: (a) 51840 273581 = 1418243x040. (b) 2x99561 = [3(523 + x)]². (c) 2784x = x 5569. %3D %3D
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
Question
6a,b,c
![(b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of N
4. For any n > 1. prove that there exists a prime with at least n of its digits equal to 0.
[Hint: Consider the arithmetic progression 10"+1k +1 for k = 1, 2, ....]
M = (100a2+ 10a1 + ao) – (100a5 +10a4+ a3)
6. Show that 2" divides an integer N if and only if 2" divides the number made up of the
5. Find the values of n > 1 for which 1!+ 2! + 3! + ·. .+n! is a perfect square.
7. Let N = am 10m + + a2102 + a¡ 10+ ao, where 0<ak <9, be the decimal expan-
(a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer
written in the base 9.
(c) Is the integer (447836)9 divisible by 3 and 8?
6. Working modulo 9 or 11, find the missing digits in the calculations below:
(a) 51840 273581 = 1418243x040.
(b) 2x99561 = [3(523 + x)]².
(c) 2784x =x 5569.
(d) 512 1x53125 = 1000000000.
7. Establish the following divisibility criteria:
(a) An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8.
(b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.
(c) An integer is divisible by 4 if and only if the number formed by its tens and u
digits is divisible by 4.
[Hint: 10 = 0 (mod 4) for k > 2.]
(d) An integer is divisible by 5 if and only if its units digit is 0 or 5.
8. For any integer a, show that a?
9. Find the remainder when 44444444 is divided by 9.
[Hint: Observe that 2 =-1 (mod 9).]
10. Prove that no integer whose digits add up to 15 can be a square or a cube.
[Hint: For any a, a’ = 0, 1, or 8 (mod 9).]
11. Assuming that 495 divides 273x49y5, obtain the digits x and
12. Determine the last three digits of the number 7999
[Hint: 74n = (1+400)" = 1 +400n (mod 1000).]
3. If t, denotes the nth triangular number, show that tn+2k = tn (mod k); hence t and t
- a +7 ends in one of the digits 3, 7, or 9.
y.
must have the same last digit.
[Hint: Problem 2(a).]
last n digits of N.
[Hint: 10k = 2k5k =0 (mod 2") for k > n.]
%3D
sion of a positive integer N.
+ (100as + 10a7 + a6) – . ..
[Hint: If n is even, then 10n = 1, 10n+1 = 10, 103n+2
1 103n+1
=100 (m
-10. 103n+2
100
odd then 103n =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F39c12f77-fd5e-41ae-b73a-621f9f2bab30%2Ffe0228d7-0ec9-4221-b9a3-10c0e0a91a80%2F5r9ahcr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Give criteria for the divisibility of N by 3 and 8 that depend on the digits of N
4. For any n > 1. prove that there exists a prime with at least n of its digits equal to 0.
[Hint: Consider the arithmetic progression 10"+1k +1 for k = 1, 2, ....]
M = (100a2+ 10a1 + ao) – (100a5 +10a4+ a3)
6. Show that 2" divides an integer N if and only if 2" divides the number made up of the
5. Find the values of n > 1 for which 1!+ 2! + 3! + ·. .+n! is a perfect square.
7. Let N = am 10m + + a2102 + a¡ 10+ ao, where 0<ak <9, be the decimal expan-
(a) Prove that 7, 11, and 13 all divide N if and only if 7, 11, and 13 divide the integer
written in the base 9.
(c) Is the integer (447836)9 divisible by 3 and 8?
6. Working modulo 9 or 11, find the missing digits in the calculations below:
(a) 51840 273581 = 1418243x040.
(b) 2x99561 = [3(523 + x)]².
(c) 2784x =x 5569.
(d) 512 1x53125 = 1000000000.
7. Establish the following divisibility criteria:
(a) An integer is divisible by 2 if and only if its units digit is 0, 2, 4, 6, or 8.
(b) An integer is divisible by 3 if and only if the sum of its digits is divisible by 3.
(c) An integer is divisible by 4 if and only if the number formed by its tens and u
digits is divisible by 4.
[Hint: 10 = 0 (mod 4) for k > 2.]
(d) An integer is divisible by 5 if and only if its units digit is 0 or 5.
8. For any integer a, show that a?
9. Find the remainder when 44444444 is divided by 9.
[Hint: Observe that 2 =-1 (mod 9).]
10. Prove that no integer whose digits add up to 15 can be a square or a cube.
[Hint: For any a, a’ = 0, 1, or 8 (mod 9).]
11. Assuming that 495 divides 273x49y5, obtain the digits x and
12. Determine the last three digits of the number 7999
[Hint: 74n = (1+400)" = 1 +400n (mod 1000).]
3. If t, denotes the nth triangular number, show that tn+2k = tn (mod k); hence t and t
- a +7 ends in one of the digits 3, 7, or 9.
y.
must have the same last digit.
[Hint: Problem 2(a).]
last n digits of N.
[Hint: 10k = 2k5k =0 (mod 2") for k > n.]
%3D
sion of a positive integer N.
+ (100as + 10a7 + a6) – . ..
[Hint: If n is even, then 10n = 1, 10n+1 = 10, 103n+2
1 103n+1
=100 (m
-10. 103n+2
100
odd then 103n =
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