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Find all solutions, if any, to the system of congruences x

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 26E Chapter 4.4

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 26E

Problem 26E

Find all solutions, if any, to the system of congruences x ≡ 5 (mod 6), x = 3 (mod 10), and x = 8 (mod 15).

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NUTR 2360 Study Guide for Exam 2 – Spring 2016 (This is not an all-­‐inclusive study guide, and this is not intended to be your only method of studying. Study your book, lecture slides and notes taken during class, and use this study guide as a checkpoint to find out which areas you need to study further. ) Carbohydrates Understand the differences between type 1, type 2, and gestational diabetes. What are the causes Treatments Consequences -­‐Type 1 diabetes: NO insulin is produced or released. *causes: Genetic component: non Hispanics whites are at greater risk. Environmental links: born to mom with gestational diabetes. Breastfeeding decreases risk of T1DM. *treatments: control blood sugar (exogenous insulin required.) carbohydrate counting. -­‐Type 2 diabetes: peripheral cells become resistant to insulin. Cells don’t respond to signal of insulin release. B-­‐islet cells become damaged. *causes: primarily environmental: high intake of sugars, obesity, increased risk if mom had gestational diabetes, breastfeeding reduc ed risk. Genetic: native Americans, African Americans, Hispanics at greater risk. *treatment: control blood sugar via low diet, increase exercise, medication, exogenous insulin. -­‐Gestational Diabetes: occurs during pregnancy *causes: high blood glucose levels during pregnancy. Glucose starts to act as a teratogen. *treatment: blood sugar MUST be controlled, insulin medication to reduce sugar absorption, low sugar diet. -­‐CONSEQUENCES FOR ALL DIABETES *damage to large blood vessels leads to: increased stroke risk, high blood pressure and increase risk of heart attack, blocked arteries in legs, reduced blood flow. *damage to the small blood vessels leads to: bleeding in retina, leading to blindness, excretion of protein in u rine and kidney failure, nerve damage that causes numbness and pain, numbness that affect balance and increases the risk of accidental injury. Know the levels of blood glucose used to diagnose diabetes (fasti ng plasma glucose test). -­‐99 and below glucose level *NORMAL/HEALTHY -­‐100-­‐125 *PRE DIABETES -­‐equal to or > than 126 *DIABETES. Protein Presence of which element makes proteins unique among the macronutrients -­‐NITROGEN What is deamination Transamination -­‐Deamination: Removing and excreting an amino group from Amino Acid. *Process: The amino group mov es to the liver and incorporated into the urea and excreted. -­‐Transamination: The process of how amino acids are synthesized. Amino acids are transferred. *process: From existing amino acid to carbon skeleton. Forms a NON -­‐ ESSENTIAL amino acid. Define essential, non-­‐essential, and conditionally essential amino acids. -­‐Essential Amino Acid: Body can’t produce, needs to be taken in from the diet. “indispensable” -­‐ ABSOLUTELY NECESSARY. -­‐Non Essential Amino Acid: Body can produce. “dispensable” -­‐ can be replaced/can do without -­‐Conditionally Essential Amino Acid: can’t get enough of. Yet, Essential during infancy, disease or trauma. *EXAMPLE: phenylketonuria(PKU) -­‐ Disorder where your body lacks a particular enzyme required to convert phenylalanine to ty rosine. Making tyrosine an essential amino acid. Must have a PARTICULAR protein diet. Describe the structure of amino acids. -­‐contains: Nitrogen group (the amino,) Acid group (carboxyl group,) hydrogen and a side chain (R.) Describe the different levels of protein structure. -­‐Primary Structure: Just a simple polypeptide chain. The order of the amino acids determines the 3D shape. Linear sequence. -­‐secondary structure: Bonding between nearby amino acids form H -­‐H. spiral shape. Has specific shape stabilized by hydrogen and sulfur bonds. -­‐Tertiary structure: has S -­‐S bonds, forms 3D shape, Determines the shape and overall function. -­‐Quaternary structure: subunits are linked together to form larger units. Describe the process of protein synthesis, including transcription, translation, and bonding between amino acids. -­‐Protein synthesis: Proteins are synthesized based on DNA template. Synthesis of protein determined through gene expression. -­‐From DNA to RNA: Transcription (copied) -­‐From RNA to Protein: Translation (ribosomes create proteins) -­‐Transcription: DNA becomes accessible, genetic code is copied and mRNA is synthesized then travels out of nucleus. -­‐Translation: mRNA is read by ribosomes, genetic code is paired with specific amino acids with specific mRNA sequences, amino acids become joined by peptide bonds. -­‐Bonds: peptide bonds and condensation reactions. Dipeptides: links TWO amino acids Polypeptides: MANY amino acids linked together. What can cause protein denaturation -­‐Altering a proteins 3D structure. Where does protein digestion begin What 2 things are responsible for protein digestion in the stomach -­‐Protein digestion begins in the STOMACH -­‐Hydrochloric acid (HCL) and Pepsin. *hydrochloric acid: starts with secretion of HCL *pepsin: breaks the polypeptide chains into shorter chains through hydrolysis What enzyme is involved in protein digestion in the intestine Where does it come from -­‐Pancreatic proteases enzyme. It comes from the pancreas. Understand how proteins are absorbed and where they go after absorption. Are whole proteins absorbed intact -­‐In the small intestine, peptides and amino acids are absorbed, in the enterocytes the peptides break down into amino acids and sent to the liver after they are release d into the blood stream. -­‐Intact proteins are not absorbed. Can protein be stored What is the amino acid pool What are the inputs and outputs -­‐There is no storage form of protein for the body. Instead, they are broken down. -­‐Amino Acid Pool: allows cells to respond to environmental changes. Cells require a pool of essential amino acids for the synthesis of body proteins. Because proteins are not stored, there is a constant “protein turnover” which is the breaking down, rebuilding, and repair of body and cellular proteins. -­‐INPUTS: Breakdown of cellular proteins (tissue), dietary protein -­‐OUTPUTS: synthesis of cellular proteins, synthesis of non protein Nitrogen containing compounds, energy production and glucose production. What are the functions of proteins within the body How do proteins help maintain fluid balance How much energy comes from protein (kcal/g) -­‐4kcal/g of energy -­‐maintaining fluid balance: two important proteins: albumin and globulin. Maintains fluid balance between blood and tissue space. (1) blood proteins draw fluid forced into interstitial spaces by Blood pressure into capillary bed. (2) without sufficient protein, edema can occur. -­‐produces vital body structures: Actin, myosin, collagen (key structural proteins) -­‐contribute to acid base balance: proteins act as BUFFERS -­‐ that help maintain acid base balance in a narrow range. They have negative charges with act with hydrogen charges. Helps with maintaining regular pH balance. -­‐forms hormones, enzymes and neurotran smitters: hormones act as messengers and controls metabolic rate and glucose taken up in bloodstream. Enzymes facilitate chemical reactions. Neurotransmitters are released by nerves. (insulin amylase and serotonin are examples.) -­‐contribute to immune function: antibody proteins. They bind to foreign proteins (antigens) that invades the body and prevents their attack on target cells. -­‐transporting nutrients: (examples: albumin and hemoglobin.) they carry nutrients through the bloodstream to cells and acros s the cell membrane to sites of action. -­‐forming glucose: gluconeogenesis-­‐if carbohydrate intake is inadequate to maintain blood glucose levels, the liver is forced to make glucose from the amino acids present in body tissues. How are plant proteins diff erent from animal proteins -­‐ When excess protein is primarily from a high intake of animal proteins, the overall diet is likely to be low in plant-based foods and consequently low in fiber, some vitamins (vitamins C and E and folate), minerals (magnesiu m and potassium), and beneficial phytochemicals . What is the difference between complete and incomplete proteins For what diet is this information especially important -­‐Complete proteins: ALL ESSENTIAL amino acids are present in adequate amounts for you r body. Source: Animal protein. -­‐Incomplete proteins: Inadequate amounts of one or more essential amino acids needed for your body. -­‐This is important to vegetarian diets. Vegetarians survive with complementary proteins which combines incomplete proteins so a person can get all of the essential amino acids they need. Which amino acids are limiting in grains, nuts & seeds and legumes -­‐GRAINS: limiting amino acids -­‐ Lysine -­‐NUTS AND SEEDS: limiting amino acids -­‐ lysine -­‐LEGUMES-­‐ limiting amino acids-­‐ methionine and tryptophan Which nutrients are of concern in vegetarian/vegan diets -­‐b12, calcium, iron, zinc, vitamin D, high quality protein, riboflavin, omega 3 fatty acids. What are some advantages of vegetarian diets -­‐lower risk of: CVD, obesity, high blood pressure (HTN) and cancer. -­‐higher consumption of: fiber, potassium, magnesium, folate and antioxidants. -­‐lower consumption of: cholesterol and saturated fat. What is the RDA for protein -­‐ 0.8g/kg What is nitrogen balance D escribe positive and negative balance and give examples. -­‐Nitrogen balance: a method to determine protein needs. -­‐Positive balance: when protein intake is greater than losses. EX: periods of recovery *input: building tissues, high dietary intake *output: urinary nitrogen excretion -­‐Negative balance: When protein intake is less than loses. EX: somebody with AIDS *input: low dietary intake *output: tissue breakdown, urinary nitrogen excretion What is Kwashiorkor Marasmus (types of PEM-­‐ protein energy malnutri tion) -­‐Kwashiorkor: pure protein deficiency, often affiliated with infection or disease. *characteristics: edema, mild to moderate weight lose, maintenance of some muscle and subcutaneous fat, growth impairment, rapid onsets and fatty liver. -­‐Marasmus: Overall energy deficiency, severe energy and protein deficit. *characteristics: severe weight loss, wasting of muscle and body fat, severe growth impairments, develops gradually. What are some problems with consuming too much protein -­‐Increased calcium excretion in urine -­‐increased fluid excretion in urine -­‐burden on the kidneys and the liver Lipids What is the primary storage form of lipids -­‐triglycerides Know the structures of triglycerides, diglycerides, and monoglycerides. -­‐Triglycerides: 3 fatty acids attached to a glycerol which serves as the backbone of fatty acids. -­‐Diglycerides: 2 fatty acids attached to a glycerol -­‐monoglycerides: 1 fatty acid attached to a glycerol Define esterification, de -­‐esterification, re-­‐esterification. -­‐Esterification: the process of attaching fatty acids to glycerol (ester bond) -­‐De-­‐esterification: the process of removing a fatty acid from a glycerol -­‐Re-­‐esterification: reattaching a fatty acid to a glycerol How do fatty acids differ from each other What determines their stability and state of matter at room temperature -­‐yes fatty acids do differ from each other. They can be classified differently by chain length, saturation and hydrogen position. -­‐The saturation of a fatty acid determines its state of matter at room temperature and stability. Be able to classify fatty acids based on chain length, degree of saturation, and hydrogen position. Be ready to do this based on images. -­‐chain length: number of carbon present *long chain: 12 or more carbon *medium chain: 6-­‐10 carbon *short chain: less than 6 carbon -­‐Saturation: number of hydrogens present *Saturated Fatty Acids: carbons are saturated with hydrogens (no double bonds) *unsaturated Fatty Acids: hydrogens are replaced carbon -­‐Hydrogen Position *Cis: SAME, Hydrogens are on the same side of the chain *trans: opposite, Hydrogens are on the opposite side of the chain Know what monounsaturated, polyunsaturated, saturated and trans fatty acids are/look like. Know the major food sources of each. -­‐monounsaturated: One C=C bond, Has a bent chain, LIQUID at room temperature and SOLID in the fridge. *plants: olive oil, canola oil, avocados, nuts *animals: chicken, turkey -­‐Polyunsaturated: Two or more C=C bonds, LIQUID at room temp erature, *plants: corn, sunflower, canola oils *animals: fish -­‐saturated: tightly packed, SOLID at room temperature, stable to heat and air, straight chain *animals: butter and lard *plants: coconut and palm oil How does hydrogenation affect the structure of fatty acids -­‐Hydrogenation: hydrogen bombards liquid oils to make saturated fatty acids. -­‐it changes the double bond configuration and makes it even more solid. What are the essential fatty acids Be able to recognize molecules from each class of essential fatty acid. What are dietary sources -­‐Essential fatty acids: required for normal functioning. Have to get from diet and can’t be created by body. -­‐Omega 6 Fatty acids: linoleic acid (LA), gamma linolenic acid (GLA), arachidonic acid (AA) -­‐Omega 3 Fatty acids: Alpha linolenic acid (ALA), eicosapentaenoic acid (EPA), Docosahexaenoic acid (DHA) -­‐Dietary sources: Plants/phytoplankton are the primary sources. (ALA) and (LA) Animals synthesize the remaining: EPA, DHA, AA from basi c molecules. What are some of the functions of essential fats in the body What are they converted to How do they influence cell communication and inflammation in the body -­‐functions: Cell membrane components-­‐ affect flexibility and fluidity vision (DHA)-­‐ Retinal development nervous system (DHA, AA) -­‐ regulates nerve transmission and communication -­‐Cell to cell communication: modulate inflammation, regulate blood pressure, regulate blood clotting. -­‐inflammation: Omega 6’s is highly inflammatory while omega 3’s is less inflammatory. -­‐Omega 3: Men-­‐ 1.6g/day, women-­‐ 1.1g/day -­‐Omega 6: men-­‐14-­‐17g/day, women-­‐11-­‐12g/day What are phospholipids Structure Food sources Functions Are they made in the body -­‐Phospholipids are AMPHIPATHIC m eaning they are hydrophobic and hydrophilic. They are the components of cell membranes and breaks down fats -­‐Structure: similar to triglycerides except one fatty acid is replaced with a phosphate that contains phosphorus and has a nitrogen attached. -­‐Phospholipids are synthesized in the body -­‐food source: egg yolks, soy, wheat germ and peanuts. What are sterols Structure Food sources Functions Are they made in the body -­‐Sterols are a type of steroid such as cholesterol, steroid hormones and vitami n D. -­‐Structure: Ring -­‐Functions: synthesizes compounds and makes compounds such as hormones. Also makes bile which emulsifies fats to be digested normally. Forms cell membranes and allows fat soluble substances to move in and out. Synthesis of bile acid s. -­‐Cholesterol is made in the liver. Synthesized in the body -­‐food sources: animal origin. What is the role of bile in lipid digestion What is the name of the structure created by bile and fat -­‐in lipid digestion, bile emulsifies fat. What is the role of pancreatic lipases Triglycerides are broken mostly down into MONOGLYCERIDES and FREE FATTY ACIDS before absorption. Be able to describe the process of lipid digestion, absorption, and transport. From bile emulsification to transport to the tissues. Know the lipoproteins, how they fit into this process, and their functions. How and where are they formed, what do they contain, what are their functions -­‐Lipid Digestion process: *Mouth-­‐ Lingual lipase is secreted. Breaks down triglycerides. Little to no FAT is digested. *stomach-­‐ gastric lipase is secreted. Breaks triglycerides into monoglycerides, diglycerides and free fatty acids. Little to no FAT is digested. *Liver-­‐ produces bile, which becomes stored and then released by the gallbladder. It empties into the small intestine. Bile emulsifies that fat forming micelles: lipid droplets formed from bile, dietary fatty acids and sterols. *Pancreas-­‐ secretes pancreatic lipase, phospholipase and cholesterol and emptie s into the small intestine. Pancreatic lipase digests triglycerides. Phospholipase digests phospholipids. Cholesterol esterase digests cholesterol. *small intestine-­‐ FAT IS DIGESTED AND ABSORBED IN THE DUODENUM AND JEJUNUM. Bile is reabsorbed in the ileum. Makes lipid accessible to lipases. Bile emulsifies fat. -­‐Lipid Digestion Lipases *triglycerides-­‐ broken down into monoglycerides and free fatty acids. *phospholipids-­‐ broken down into free fatty acids, glycerol and phosphoric acid *cholesterol esters-­‐ broken down into cholesterol and free fatty acids. -­‐Liquid absorption: (1) large fat droplets enter the small intestine. (2) Bile emulsifies the fat into smaller particles. (3) lipase break s down fat into fatty acids and monoglycerides. (4) Monoglycerides and fatty acids are absorbed as micelles through the brush border and then re formed into triglycerides. (5) short and medium chain fatty acids enter the bloodstream. (6) triglyceri des combined with cholesterol, protein, and phospholipids to form chylomicrons, which enter the lymphatic system. -­‐Lipid transport to the body *chylomicrons-­‐ made in ENTEROCYTES. Transports DIETARY FATS, FAT SOLUBLE VITAMINS. From small intestines to the tissues and eventually re absorbed into the liver. *inside enterocytes-­‐ short and medium chains move into the portal vein. Long chains packaged into chylomicrons and secreted into the lymphatic system. *Very low density lipoproteins (VLDL)-­‐ made by the LIVER. Transports triglycerides from liver to tissues. Contains cholesterol. When fat is delivered, it transforms into intermediate density lipoprotein. *low density lipoproteins (LDL)-­‐ formed from IDL/VLDL. Transports CHOLESTEROL into the cells and eventually gets absorbed into the liver. BIGGEST CONTRIBUTOR TO PLAQUE FORMATION *high density lipoproteins (HDL) -­‐ THESE ARE THE GOOD FATS!!!made by the LIVER, picks up cholesterol from tissues. Returns the cholesterol to the liver for recycling or removal. “reverse cholesterol transport.” The higher levels reduce CVD risk. What is reverse cholesterol transport -­‐the net movement of cholesterol from peripheral tissues BACK to the liver for removal or recycling. Be able to identify modifiable and non -­‐modifiable risk f actors for developing cardiovascular disease (CVD). -­‐modifiable: blood triglyceride and cholesterol levels, hypertension, smoking, physical inactivity, obesity and diabetes. -­‐non modifiable: age, gender, race and genetics What are the three primary contributors to CVD -­‐inflammation *contributes to atherosclerotic plaque formation -­‐high LDL cholesterol *contributes to atherosclerotic plaque formation -­‐hypertension *contributes to damage of blood vessels What is atherosclerosis How does a plaque form -­‐Atherosclerosis is hardening of the arteries. -­‐plaque forms from foam cells. Which fats contribute the most to cardiovascular disease risk -­‐high fats diets, mostly rich in saturated and trans fats increase the risk. Know the normal and abnormal levels for tota l, HDL, and LDL cholesterol. -­‐Total *desirable: <200. Borderline high: 200 -­‐239. High: equal to or more than 240. -­‐LDL *optimal: <100. Near optimal: 100-­‐129. Borderline high: 130-­‐159. High: 160-­‐189. Very high: equal to or greater than 190. -­‐HDL *low: <40. High: equal to o r greater than 60. Describe what occurs during a heart attack. -­‐(1) plaques break free from the blood vessel walls. ( 2) stimulates clot formation. (3) blood flow is blocked. (4) tissue dies from lack of blood flow causing myocardial infraction in heart and causes cerebral vascular accidents in the brain. What are the recommended levels of essential fatty acid intake -­‐omega 3: -­‐Men: 1.6g/day women:1.1g/day -­‐Omega 6: -­‐Men:14-­‐17g/day women:11-­‐12g/day

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Chapter 4.4, Problem 26E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

This full solution covers the following key subjects: mod, any, Find, Congruences, Solutions. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. The full step-by-step solution to problem: 26E from chapter: 4.4 was answered by , our top Math solution expert on 06/21/17, 07:45AM. The answer to “Find all solutions, if any, to the system of congruences x ? 5 (mod 6), x = 3 (mod 10), and x = 8 (mod 15).” is broken down into a number of easy to follow steps, and 26 words. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since the solution to 26E from 4.4 chapter was answered, more than 321 students have viewed the full step-by-step answer.

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Find all solutions, if any, to the system of congruences x